R. TRACY  CRAWFORD 


Astronomy  Library 


AN  INTRODUCTION  TO 
CELESTIAL  MECHANICS 


THE  MACMILLAN   COMPANY 

NEW  YORK          BOSTON          CHICAGO 
DALLAS          SAN    FRANCISCO 

MACMILLAN  &  CO.,  LIMITED 

LONDON      BOMBAY       CALCUTTA 
MELBOURNE 

THE  MACMILLAN  CO.  OF  CANADA,  LTD. 

TORONTO 


AN  INTRODUCTION 

TO 

CELESTIAL   MECHANICS 


BY 


FOREST  RAY  MOULTON,  PH.D. 

PROFESSOR   OF   ASTRONOMY   IN   THE    UNIVERSITY    OF    CHICAGO 
RESEARCH    ASSOCIATE    OF   THE    CARNEGIE   INSTITUTION    OF   WASHINGTON 


SECOND  REVISED  EDITION 


Neto  ¥0rfc 
THE  MACMILLAN  COMPANY 

LONDON:   MACMILLAN  &  CO.,  LTD. 
1914 

All  rights  reserved 


COPYRIGHT,   1914 
BY  THE  MACMILLAN  COMPANY 

Set  up  and  electrotyped.    Published  April,  1914 


ASTRONOMY  DEFT 


PREFACE  TO  FIRST  EDITION. 

AN  attempt  has  been  made  in  this  volume  to  give  a  somewhat 
satisfactory  account  of  many  parts  of  Celestial  Mechanics 
rather  than  an  exhaustive  treatment  of  any  special  part.  The  aim 
has  been  to  present  the  work  so  as  to  attain  logical  sequence,  to 
make  it  progressively  more  difficult,  and  to  give  the  various  subjects 
the  relative  prominence  which  their  scientific  and  educational 
importance  deserves.  In  short,  the  aim  has  been  to  prepare  such 
a  book  that  one  who  has  had  the  necessary  mathematical  training 
may  obtain  from  it  in  a  relatively  short  time  and  by  the  easiest 
steps  a  sufficiently  broad  and  just  view  of  the  whole  subject  to 
enable  him  to  stop  with  much  of  real  value  in  his  possession,  or  to 
pursue  to  the  best  advantage  any  particular  portion  he  may  choose. 

In  carrying  out  the  plan  of  this  work  it  has  been  necessary  to  give 
an  introduction  to  the  Problem  of  Three  Bodies.  This  is  not  only 
one  of  the  justly  celebrated  problems  of  Celestial  Mechanics,  but  it 
has  become  of  special  interest  in  recent  times  through  the  researches 
of  Hill,  Poincare,  and  Darwin.  The  theory  of  absolute  pertur- 
bations is  the  central  subject  in  mathematical  Astronomy,  and 
such  a  work  as  this  would  be  inexcusably  deficient  if  it  did  not 
give  this  theory  a  prominent  place.  A  chapter  has  been  devoted 
to  geometrical  considerations  on  perturbations.  Although  these 
methods  are  of  almost  no  use  in  computing,  yet  they  furnish  in  a 
simple  manner  a  clear  insight  into  the  nature  of  the  problem,  and 
are  of  the  highest  value  to  beginners.  The  fundamental  principles 
of  the  analytical  methods  have  been  given  with  considerable 
completeness,  but  many  of  the  details  in  developing  the  formulas 
have  been  omitted  in  order  that  the  size  of  the  book  might  not 
defeat  the  object  for  which  it  has  been  prepared.  The  theory  of 
orbits  has  not  been  given  the  unduly  prominent  position  which  it 
has  occupied  in  this  country,  doubtless  due  to  the  influence  of 
Watson's  excellent  treatise  on  this  subject. 

The  method  of  treatment  has  been  to  state  all  problems  in 
advance  and,  where  the  transformations  are  long,  to  give  an 
outline  of  the  steps  which  are  to  be  made.  The  expression  "order 
of  small  quantities"  has  not  been  used  except  when  applied  to 
power  series  in  explicit  parameters,  thus  giving  the  work  all  the 

M572515 


VI  PREFACE    TO    FIRST   EDITION. 

definiteness  and  simplicity  which  are  characteristic  of  operations 
with  power  series.  This  is  exemplified  particularly  in  the  chapter 
on  perturbations.  Care  has  been  taken  to  make  note  at  all 
places  where  assumptions  have  been  'introduced  or  unjustified 
methods  employed,  for  it  is  only  by  seeing  where  the  points  of 
possible  weakness  are  that  improvements  can  be  made.  The 
frequent  references  throughout  the  text  and  the  bibliographies  at 
the  ends  of  the  chapters,  though  by  no  means  exhaustive,  are 
sufficient  to  direct  one  in  further  reading  to  important  sources 
of  information. 

This  volume  is  the  outgrowth  of  a  course  of  lectures  given 
annually  by  the  author  at  the  University  of  Chicago  during  the 
last  six  years.  These  lectures  have  been  open  to  senior  college 
students  and  to  graduate  students  who  have  not  had  the  equivalent 
of  this  work.  They  have  been  taken  by  students  of  Astronomy, 
by  many  making  Mathematics  their  major  work,  and  by  some  who, 
though  specializing  in  quite  distinct  lines,  have  desired  to  get  an 
idea  of  the  processes  by  means  of  which  astronomers  interpret 
and  predict  celestial  phenomena.  Thus  they  have  served  to 
give  many  an  idea  of  the  methods  of  investigation  and  the  results 
attained  in  Celestial  Mechanics,  and  have  prepared  some  for  a 
detailed  study  extending  into  the  various  branches  of  modern 
investigations.  The  object  of  the  work,  the  subjects  covered,  and 
the  methods  of  treatment  seem  to  have  been  amply  justified  by 
this  experience. 

Mr.  A.  C.  Lunn,  M.A.,  has  read  the  entire  manuscript  with  great 
care  and  a  thorough  insight  into  the  subjects  treated.  His  nu- 
merous corrections  and  suggestions  have  added  greatly  to  the 
accuracy  and  the  method  of  treatment  in  many  places.  Professor 
Ormond  Stone  has  read  the  proofs  of  the  first  four  chapters  and 
the  sixth.  His  experience  as  an  investigator  and  as  a  teacher  has 
made  his  criticisms  and  suggestions  invaluable.  Mr.  W.  O.  Beal, 
M.A.,  has  read  the  proofs  of  the  whole  book  with  great  attention 
and  he  is  responsible  for  many  improvements.  The  author  desires 
to  express  his  sincerest  thanks  to  all  these  gentlemen  for  the 
willingness  and  the  effectiveness  with  which  they  have  devoted  so 
much  of  their  time  to  this  work. 

F.  R.  MOULTON. 
CHICAGO,  July,  1902. 


PREFACE  TO  SECOND  EDITION. 

THE  necessity  for  a  new  edition  of  this  work  has  given  the 
opportunity  of  thoroughly  revising  it.  The  general  plan 
which  has  been  followed  is  the  same  as  that  of  the  first  edition, 
because  it  was  found  that  it  satisfies  a  real  need  not  only  in  this 
country,  for  whose  students  it  was  primarily  written,  but  also  in 
Europe.  In  spite  of  all  temptations  its  elementary  character  has 
been  preserved,  and  it  has  not  been  greatly  enlarged.  Very 
many  improvements  have  been  made,  partly  on  the  suggestion  of 
numerous  astronomers  and  mathematicians,  and  it  is  hoped  that 
it  will  be  found  more  worthy  of  the  favor  with  which  it  has  so  far 
been  received. 

The  most  important  single  change  is  in  the  discussion  of  the 
methods  of  determining  orbits.  This  subject  logically  follows  the 
Problem  of  Two  Bodies,  and  it  is  much  more  elementary  in  char- 
acter than  the  Problem  of  Three  Bodies  and  the  Theory  of  Per- 
turbations. For  these  reasons  it  was  placed  in  chapter  VI.  The 
subject  matter  has  also  been  very  much  changed.  The  methods 
of  Laplace  and  Gauss,  on  which  all  other  methods  of  general  applic- 
ability are  more  or  less  directly  based,  are  both  given.  The 
standard  modes  of  presentation  have  not  been  followed  because, 
however  well  they  may  be  adapted  to  practice,  they  are  not  noted 
for  mathematical  clarity.  Besides,  there  is  no  lack  of  excellent 
works  giving  details  in  the  original  forms  and  models  of  com- 
putation. The  other  changes  and  additions  of  importance  are 
in  the  chapters  on  the  Problem  of  Two  Bodies,  the  Problem  of 
Three  Bodies,  and  in  that  on  Geometrical  Consideration  of  Per- 
turbations. 

It  is  a  pleasure  to  make  special  acknowledgment  of  assistance 
to  my  colleague  Professor  W.  D.  MacMillan  and  to  Mr.  L.  A. 
Hopkins  who  have  read  the  entire  proofs  not  only  once  but  several 
times,  and  who  have  made  important  suggestions  and  have  pointed 
out  many  defects  that  otherwise  would  have  escaped  notice. 
They  are  largely  responsible  for  whatever  excellence  of  form  the 
book  may  possess. 

F.  R.  MOULTON. 

CHICAGO,  January,  1914. 


vn 


TABLE  OF  CONTENTS. 

CHAPTER  I. 
FUNDAMENTAL  PRINCIPLES  AND  DEFINITIONS. 

ART.  PAGE 

1 .  Elements  and  laws 

2.  Problems  treated 

3.  Enumeration  of  the  principal  elements      ...... 

4.  Enumeration  of  principles  and  laws 3 

5.  Nature  of  the  laws  of  motion 

6.  Remarks  on  the  first  law  of  motion    .               4 

7.  Remarks  on  the  second  law  of  motion 4 

8.  Remarks  on  the  third  law  of  motion  .        .       .        .        .        .        .  6 

DEFINITIONS  AND  GENERAL  EQUATIONS 8 

9.  Rectilinear  motion,  speed,  velocity     .......  8 

10.  Acceleration  in  rectilinear  motion 9 

11.  Speed  and  velocities  in  curvilinear  motion         .        .        ...        .  10 

12.  Acceleration  in  curvilinear  motion 11 

13.  Velocity  along  and  perpendicular  to  the  radius  vector    .        .        .  12 

14.  The  components  of  acceleration 13 

15.  Application  to  a  particle  moving  in  a  circle 14 

16.  The  areal  velocity  . 15 

17.  Application  to  motion  in  an  ellipse 16 

Problems  on  velocity  and  acceleration 17 

18.  Center  of  mass  of  n  equal  particles 19 

19.  Center  of  mass  of  unequal  particles 20 

20.  The  center  of  gravity 22 

21.  Center  of  mass  of  a  continuous  body 24 

22.  Planes  and  axes  of  symmetry 26 

23.  Application  to  a  non-homogeneous  cube 26 

24.  Application  to  the  octant  of  a  sphere 27 

Problems  on  center  of  mass 28 

HISTORICAL  SKETCH  FROM  ANCIENT  TIMES  TO  NEWTON. 

25.  The  two  divisions  of  the  history 29 

26.  Formal  astronomy 30 

27.  Dynamical  astronomy 33 

Bibliography 35 

CHAPTER  II. 

RECTILINEAR  MOTION. 

THE  MOTION  OF  FALLING  PARTICLES. 

29.  The  differential  equations  of  motion 36 

30.  Case  of  constant  force 37 

ix 


X  TABLE    OF   CONTENTS. 

ART.  PAGE 

31.  Attractive  force  varying  directly  as  the  distance      ....  38 
Problems  on  rectilinear  motion    .        . 40 

32.  Solution  of  linear  equations  by  exponentials              .        .        .        .  41 

33.  Attractive  force  varying  inversely  as  the  square  of  the  distance     .  43 

34.  The  height  of  projection 45 

35.  The  velocity  from  infinity 45 

36.  Application  to  the  escape  of  atmospheres 46 

37.  The  force  proportional  to  the  velocity 49 

38.  The  force  proportional  to  the  square  of  the  velocity        ...  53 
Problems  on  linear  differential  equations 55 

39.  Parabolic  motion .56 

Problems  on  parabolic  motion 58 

THE  HEAT  OF  THE  SUN. 

40.  Work  and  energy                           / 59 

41.  Computation  of  work 59 

42.  The  temperature  of  meteors 61 

43.  The  meteoric  theory  of  the  sun's  heat 62 

44.  Helmholtz's  contraction  theory 63 

Problems  on  heat  of  sun 66 

Historical  sketch  and  bibliography 67 

CHAPTER  III. 

CENTRAL   FORCES. 

45.  Central  force 69 

46.  The  law  of  areas 69 

47.  Analytical  demonstration  of  the  law  of  areas 71 

48.  Converse  of  the  theorem  of  areas 73 

49.  The  laws  of  angular  and  linear  velocity 73 

SIMULTANEOUS  DIFFERENTIAL  EQUATIONS. 

50.  The  order  of  a  system  of  simultaneous  differential  equations         .  74 

51.  Reduction  of  order 77 

Problems  on  differential  equations 78 

52.  The  vis  viva  integral 78 

EXAMPLES  WHERE  /  is  A  FUNCTION  OF  THE  COORDINATES  ALONE. 

53.  Force  varying  directly  as  the  distance 79 

54.  Differential  equation  of  the  orbit 80 

55.  Newton's  law  of  gravitation 82 

56.  Examples  of  finding  the  law  of  force 84 

THE  UNIVERSALITY  OF  NEWTON'S  LAW. 

57.  Double  star  orbits 85 

58.  Law  of  force  in  binary  stars 86 

59.  Geometrical  interpretation  of  the  second  law 88 

60.  Examples  of  conic  section  motion 89 

Problems  of  finding  law  of  force 89 

DETERMINATION  OF  THE  ORBIT  FROM  THE  LAW  OF  FORCE. 

61.  Force  varying  as  the  distance '.90 

62.  Force  varying  inversely  as  the  square  of  the  distance      ...  92 


TABLE   OF   CONTENTS.  XI 

ART.  PAGE 

63.     Force  varying  inversely  as  the  fifth  power  of  the  distance     .        .       93 

Problems  on  determining  orbits  from  law  of  force   ....       95 

Historical  sketch  and  bibliography 97 

CHAPTER   IV. 

THE  POTENTIAL  AND  ATTRACTIONS  OF  BODIES. 

65.  Solid  angles 98 

66.  The  attraction  of  a  thin  homogeneous  spherical  shell  upon  a 

particle  in  its  interior 99 

67.  The  attraction  of  a  thin  homogeneous  ellipsoidal  shell  upon  a 

particle  in  its  interior 100 

68.  The  attraction  of  a  thin  homogeneous  spherical  shell  upon  an 

exterior  particle.     Newton's  method 101 

69.  Comments  upon  Newton's  method 103 

70.  The  attraction  of  a  thin  homogeneous  spherical  shell  upon  an 

exterior  particle.     Thomson  and  Tait's  method   ....  104 

71.  Attraction  upon  a  particle  in  a  homogeneous  spherical  shell      .  106 
Problems  on  attractions  of  simple  solids 107 

72.  The   general   equations   for   the   components   of   attraction   and 

for  the  potential  when  the  attracted  particle  is  not  a  part 

of  the  attracting  mass 108 

73.  Case  where  the  attracted  particle  is  a  part  of  the  attracting  mass     110 

74.  Level  surfaces 113 

75.  The  potential  and  attraction  of  a  thin  homogeneous  circular 

disc  upon  a  particle  in  its  axis . 113 

76.  The  potential  and  attraction  of  a  thin  homogeneous  spherical 

shell  upon  an  interior  or  an  exterior  particle 114 

77.  Second  method  of  computing  the  attraction  of  a  homogeneous 

sphere 115 

Problems  on  the  potential  and  attractions  of  simple  bodies     .        .118 

78.  The  potential   and   attraction   of  a   solid   homogeneous   oblate 

spheroid  upon  a  distant  particle 119 

79.  The  potential  and  attraction  of  a  solid  homogeneous  ellipsoid 

upon  a  unit  particle  in  its  interior 122 

Problems  on  the  potential  and  attractions  of  ellipsoids  .        .        .126 

80.  The  attraction  of  a  solid  homogeneous  ellipsoid  upon  an  exterior 

particle.     Ivory's  method 127 

81.  The  attraction  of  spheroids 132 

82.  The  attraction  at  the  surfaces  of  spheroids 133 

Problems  on  Ivory's  method  and  level  surfaces        ....  137 

Historical  sketch  and  bibliography 138 

CHAPTER  V. 
THE   PROBLEM   OF  TWO   BODIES. 

83.  Equations  of  motion 140 

84.  The  motion  of  the  center  of  mass 141 


Xll  TABLE    OF   CONTENTS. 

ART.  PAGE 

85.  The  equations  for  relative  motion 142 

86.  The  integrals  of  areas 144 

87.  Problem  in  the  plane 146 

88.  The  elements  in  terms  of  the  constants  of  integration     .        .        .148 

89.  Properties  of  the  motion 149 

90.  Selection  of  units  and  determination  of  the  constant  k   .       .        .153 
Problems  on  elements  of  orbits 154 

91.  Position  in  parabolic  orbits 155 

92.  Equation  involving  two  radii  and  their  chord.     Euler's  equation     157 

93.  Position  in  elliptic  orbits 158 

94.  Geometrical  derivation  of  Kepler's  equation 159 

95.  Solution  of  Kepler's  equation 160 

96.  Differential  corrections 162 

97.  Graphical  solution  of  Kepler's  equation     .        .        .        .        .        .163 

98.  Recapitulation  of  formulas .164 

99.  The  development  of  E  in  series .        .165 

100.  The  development  of  r  and  v  in  series 169 

101.  Direct  computation  of  the  polar  coordinates 172 

102.  Position  in  hyperbolic  orbits 177 

103.  Position  in  elliptic  and  hyperbolic  orbits  when  e  is  near  unity       .      178 
Problems  on  expansions  and  positions  in  orbits        .        .        .        .181 

104.  The  heliocentric  position  in  the  ecliptic  system        .        .        .        .182 

105.  Transfer  of  the  origin  to  the  earth 185 

106.  Transformation  to  geocentric  equatorial  coordinates       .        .        .186 

107.  Direct  computation  of  the  geocentric  equatorial  coordinates        .      187 

Problems  on  transformations  of  coordinates 189 

Historical  sketch  and  bibliography 190 

CHAPTER  VI. 

THE   DETERMINATION   OF  ORBITS. 

108.  General  considerations  . 191 

109.  Intermediate  elements 192 

110.  Preparation  of  the  observations  .        . 194 

111.  Outline  of  the  Laplacian  method  of  determining  orbits  .        .        .195 

112.  Outline  of  the  Gaussian  method  of  determining  orbits    .        .        .199 
I.     THE  LAPLACIAN  METHOD  OF  DETERMINING  ORBITS. 

113.  Determination  of  the  first  and  second  derivatives  of  the  angular 

coordinates  from  three  observations 202 

114.  Determination   of  the   derivatives   from   more   than   three   ob- 

servations      205 

115.  The  approximations  in  the  determination  of  the  values  of  X,  /*,  v 

and  their  derivatives 206 

116.  Choice  of  the  origin  of  time 207 

117.  The  approximations  when  there  are  four  observations     .        .        .     208 

118.  The  fundamental  equations 211 

119.  The  equations  for  the  determination  of  r  and  p        .        .        .        .212 

120.  The  condition  for  a  unique  solution 215 


TABLE    OF    CONTENTS.  Xlll 

ART.  PAGE 

121.  Use  of  a  fourth  observation  in  case  of  a  double  solution         .        .  218 

122.  The  limits  on  m  and  M 219 

123.  Differential  corrections 220 

124.  Discussion  of  the  determinant  D 222 

125.  Reduction  of  the  determinants  Z)i  and  D 2 224 

126.  Correction  for  the  time  aberration 226 

127.  Development  of  x,  y,  and  z  in  series   . 227 

128.  Computation  of  the  higher  derivatives  of  X,  /*,  v              .        .        .  229 

129.  Improvement  of  the  values  of  x,  y,  z,  x',  y',  z' 230 

130.  The  modifications  of  Harzer  and  Leuschner 231 

II.     THE  GAUSSIAN  METHOD  OF  DETERMINING  ORBITS. 

131.  The  equation  for  P2        ... 232 

132.  The  equations  for  p\  and  p3  . 236 

133.  Improvement  of  the  solution 236 

134.  The  method  of  Gauss  for  computing  the  ratios  of  the  triangles       .  237 

135.  The  first  equation  of  Gauss  .        .      . »  -    • 238 

136.  The  second  equation  of  Gauss      .        . 240 

137.  Solution  of  (98)  and  (101) 241 

138.  Determination  of  the  elements  a,  e,  and  o>         .....  243 

139.  Second  method  of  determining  a,  e,  and  w 244 

140.  Computation  of  the  time  of  perihelion  passage 248 

141.  Direct  derivation  of  equations  defining  orbits 249 

142.  Formulas  for  computing  an  approximate  orbit 250 

Problems  on  determining  orbits 257 

Historical  sketch  and  bibliography .  258 

CHAPTER  VII. 
THE   GENERAL  INTEGRALS  OF  THE  PROBLEM  OF  n  BODIES. 

143.  The  differential  equations  of  motion 261 

144.  The  six  integrals  of  the  motion  of  the  center  of  mass      .        .        .  262 

145.  The  three  integrals  of  areas 264 

146.  The  energy  integral 267 

147.  The  question  of  new  integrals 268 

Problems  on  motion  of  center  of  mass  and  areas  integrals     «.        .  269 

148.  Transfer  of  the  origin  to  the  sun 269 

149.  Dynamical  meaning  of  the  equations 271 

150.  The  order  of  the  system  of  equations 273 

Problems  on  differential  equations  for  motion  of  n  bodies      .        .  274 

Historical  sketch  and  bibliography 275 

CHAPTER  VIII. 
THE  PROBLEM   OF  THREE   BODIES. 

151.  Problem  considered 277 

MOTION  OF  THE  INFINITESIMAL  BODY. 

152.  The  differential  equations  of  motion 278 


XIV  TABLE    OF   CONTENTS. 

ART.  PAGE 

153.  Jacobi's  integral 280 

154.  The  surfaces  of  zero  relative  velocity 281 

155.  Approximate  forms  of  the  surfaces 282 

156.  The  regions  of  real  and  imaginary  velocity 286 

157.  Method  of  computing  the  surfaces 287 

158.  Double  points  of  the  surfaces  and  particular  solutions  of  the 

problem  of  three  bodies 

Problems  on  surfaces  of  zero  relative  velocity 

159.  Tisserand's  criterion  for  the  identity  of  comets        .... 

160.  Stability  of  particular  solutions 

161.  Application  of  the  criterion  for  stability  to  the  straight  line 

solutions 300 

162.  Particular  values  of  the  constants  of  integration      ....  302 

163.  Application  to  the  gegenschein 305 

164.  Application  of  the  criterion  for  stability  to  the  equilateral  triangle 

solutions     .... 306 

Problems  on  motion  of  infinitesimal  body 308 

CASE  OF  THREE  FINITE  BODIES. 

165.  Conditions  for  circular  orbits 309 

166.  Equilateral  triangle  solutions       .        . 310 

167.  Straight  line  solutions 311 

168.  Dynamical  properties  of  the  solutions 312 

169.  General  conic  section  solutions 313 

Problems  on  particular  solutions  of  the  problem  of  three  bodies     .  318 

Historical  sketch  and  bibliography 319 

CHAPTER  IX. 

PERTURBATIONS— GEOMETRICAL  CONSIDERATIONS. 

170.  Meaning  of  perturbations 321 

171.  Variation  of  coordinates 321 

172.  Variation  of  the  elements 322 

173.  Derivation  of  the  elements  from  a  graphical  construction      .        .  323 

174.  Resolution  of  the  disturbing  force 324 

I.  EFFECTS  OF  THE  COMPONENTS  OF  THE  DISTURBING  FORCE. 

175.  Disturbing  effects  of  the  orthogonal  component       .....  325 

176.  Effects  of  the  tangential  component  upon  the  major  axis       .        .  327 

177.  Effects  of  the  tangential  component  upon  the  line  of  apsides        .  327 

178.  Effects  of  the  tangential  component  upon  the  eccentricity     .        .  328 

179.  Effects  of  the  normal  component  upon  the  major  axis    .        .        .  329 

180.  Effects  of  the  normal  component  upon  the  line  of  apsides      .        .  329 

181.  Effects  of  the  normal  component  upon  the  eccentricity  .        .        .331 

182.  Table  of  results 332 

183.  Disturbing  effects  of  a  resisting  medium 333 

184.  Perturbations  arising  from  oblateness  of  the  central  body      .        .  333 
Problems  on  perturbations 335 

II.  THE  LUNAR  THEORY. 

185.  Geometrical  resolution  of  the  disturbing  effects  of  a  third  body    .  337 


TABLE    OF    CONTENTS.  XV 

ART.  PAGE 

186.  Analytical  resolution  of  the  disturbing  effects  of  a  third  body     .  338 

187.  Perturbations  of  the  node 342 

188.  Perturbations  of  the  inclination 343 

189.  Precession  of  the  equinoxes.     Nutation 344 

190.  Resolution  of  the  disturbing  acceleration  in  the  plane  of  motion  .  345 

191.  Perturbations  of  the  major  axis 346 

192.  Perturbation  of  the  period .        .        .  348 

193.  The  annual  equation 348 

194.  The  secular  acceleration  of  the  moon's  mean  motion       .        .        .  348 

195.  The  variation 350 

196.  The  parallactic  inequality 352 

197.  The  motion  of  the  line  of  apsides 352 

198.  Secondary  effects 355 

199.  Perturbations  of  the  eccentricity 356 

200.  The  erection 359 

201.  Gauss'  method  of  computing  secular  variations        ....  360 

202.  The  long  period  inequalities 361 

Problems  on  perturbations 362 

Historical  sketch  and  bibliography 363 

CHAPTER  X. 

PERTURBATIONS— ANALYTICAL  METHOD. 

203.  Introductory  remarks 366 

204.  Illustrative  example 367 

205.  Equations  in  the  problem  of  three  bodies 372 

206.  Transformation  of  variables 374 

207.  Method  of  solution 377 

208.  Determination  of  the  constants  of  integration 381 

209.  The  terms  of  the  first  order 382 

210.  The  terms  of  the  second  order 383 

Problems  on  the  method  of  computing  perturbations      .        .        .  386 

211.  Choice  of  elements 387 

212.  Lagrange's  brackets 387 

213.  Properties  of  Lagrange's  brackets 388 

214.  Transformation  to  the  ordinary  elements 390 

215.  Method  of  direct  computation  of  Lagrange's  brackets    .        .        .391 

216.  Computation  of  [co,  ft],  [ft,  i],  [i,  co] 395 

217.  Computation  of  [K,  P] 396 

218.  Computation  of  [a,  e],  [e,  a],  [<r,  a] 397 

219.  Change  from  ft,  co,  and  a  to  ft,  TT,  and  e 400 

220.  Introduction  of  rectangular  components  of  the  disturbing  ac- 

celeration       402 

Problems  on  variation  of  elements 405 

221.  Development  of  the  perturbative  function 406 

222.  Development  of  Ri,2  in  the  mutual  inclination 407 

223.  Development  of  the  coefficients  in  powers  of  e\  and  e*    .        .        .  409 

224.  Developments  in  Fourier  series 410 


XVI  TABLE    OF    CONTENTS. 

ART.  PAGE 

225.  Periodic  variations .                                       413 

226.  Long  period  variations  .        .        . 416 

227.  Secular  variations 417 

228.  Terms  of  the  second  order  with  respect  to  the  masses     .        .        .  419 

229.  Lagrange's  treatment  of  the  secular  variations 420 

230.  Computation  of  perturbations  by  mechanical  quadratures     .        .  425 

231.  General  reflections 429 

Problems  on  the  perturbative  function 430 

Historical  sketch  and  bibliography      .        .        .        .        ,        .        .431 


INTRODUCTION  TO  CELESTIAL  MECHANICS 


CHAPTER   I. 

FUNDAMENTAL  PRINCIPLES    AND   DEFINITIONS. 

1.  Elements  and  Laws.  The  problems  of  every  science  are 
expressible  in  certain  terms  which  will  be  designated  as  elements, 
and  depend  upon  certain  principles  and  laws  for  their  solution. 
The  elements  arise  from  the  very  nature  of  the  subject  considered, 
and  are  expressed  or  implied  in  the  formulation  of  the  problems 
treated.  The  principles  and  laws  are  the  relations  which  are 
known  or  are  assumed  to  exist  among  the  various  elements. 
They  are  inductions  from  experiments,  or  deductions  from  previ- 
ously accepted  principles  and  laws,  or  simply  agreements. 

An  explicit  statement  in  the  beginning  of  the  type  of  problems 
which  will  be  treated,  and  an  enumeration  of  the  elements  which 
they  involve,  and  of  the  principles  and  laws  which  relate  to  them, 
will  lead  to  clearness  of  exposition.  In  order  to  obtain  a  com- 
plete understanding  of  the  character  of  the  conclusions  which  are 
reached,  it  would  be  necessary  to  make  a  philosophical  discussion 
of  the  reality  of  the  elements,  and  of  the  origin  and  character  of 
the  principles  and  laws.  These  questions  cannot  be  entered  into 
here  because  of  the  difficulty  and  complexity  of  metaphysical 
speculations.  It  is  not  to  be  understood  that  such  investigations 
are  not  of  value;  they  forever  lead  back  to  simpler  and  more 
undeniable  assumptions  upon  which  to  base  all  reasoning. 

The  method  of  procedure  in  this  work  will  necessarily  be  to 
accept  as  true  certain  fundamental  elements  and  laws  without 
entering  in  detail  into  the  questions  of  their  reality  or  validity. 
It  will  be  sufficient  to  consider  whether  they  are  definitions  or 
have  been  inferred  from  experience,  and  to  point  out  that  they 
have  been  abundantly  verified  in  their  applications.  They  will  be 
accepted  with  confidence,  and  their  consequences  will  be  derived, 
in  the  subjects  treated,  so  far  as  the  scope  and  limits  of  the  work 
will  allow. 


i^ATED.  [2 


A  PROBLEMS   TH^ATED. 

2.  Problems  Treated.     The  motions  of  a  material  particle  sub- 
ject to  a  central  force  of  any  sort  whatever  will  be  briefly  con- 
sidered.    It  will  be  shown  from  the  conclusions  reached  in  this  dis- 
cussion, and  from  the  observed  motions  of  the  planets  and  their 
satellites,  that  Newton's  law  of  gravitation  holds  true  in  the  solar 
system.     The  character  of  the  motion  of  the  binary  stars  shows 
that  the  probabilities  are  very  great  that  it  operates  in  them  also, 
and  that  it  may  well  be  termed  "the  law  of  universal  gravita- 
tion."    This  conclusion  is  confirmed  by  the  spectroscope,  which 
proves  that  the  familiar  chemical  elements  of  our  solar  system 
exist  in  the  stars  also. 

In  particular,  the  motions  of  two  free  homogeneous  spheres 
subject  only  to  their  mutual  attractions  and  starting  from  arbi- 
trary initial  conditions  will  be  investigated,  and  then  their  motions 
will  be  discussed  when  they  are  subject  to  disturbing  influences  of 
various  sorts.  The  essential  features  of  perturbations  arising  from 
the  action  of  a.  third  body  will  be  developed,  both  from  a  geo- 
metrical and  an  analytical  point  of  view.  There  are"  two  some- 
what different  cases.  One  is  that  in  which  the  motion  of  a  satel- 
^*A-  lite  around  a  planet  is  perturbed  by  the  sun;  and  the  second  is 
that  in  which  the  motion  of  one  planet  around  the  sun  is  per- 
1  turbed  by  another  planet. 

Another  class  of  problems  which  arises  is  the  determination  of 
the  orbits  of  unknown  bodies  from  the  observations  of  their  direc- 
tions at  different  epochs,  made  from  a  body  whose  motion  is 
known.  That  is,  the  theories  of  the  orbits  of  comets  and  plan- 
etoids will  be  based  upon  observations  of  their  apparent  positions 
made  from  the  earth.  This  incomplete  outline  of  the  questions 
to  be  treated  is  sufficient  for  the  enumeration  of  the  elements 
employed. 

3.  Enumeration  of  the  Principal  Elements.     In  the  discussion 
of  the  problems  considered  in  this  work  it  will  be  necessary  to 
employ  the  following  elements: 

(a)  Real  numbers,  and  complex  numbers  incidentally  in  the 
solution  of  certain  problems. 

(6)  Space  of  three  dimensions,  possessing  the  same  properties  in 
every  direction. 

(c)  Time  of  one  dimension,  which  will  be  taken  as  the  inde- 
pendent variable. 

(d)  Mass,  having  the  ordinary  properties  of  inertia,  etc.,  which 
are  postulated  in  elementary  Physics. 


5]  NATURE   OF   THE   LAWS   OF   MOTION.  3 

(e)   Force,  with  the  content  that  the  same  term  has  in  Physics. 

Positive  numbers  arise  in  Arithmetic,  and  positive,  negative, 
and  complex  numbers,  in  Algebra.  Space  appears  first  as  an 
essential  element  in  Geometry.  Time  appears  first  as  an  essential 
element  in  Kinematics.  Mass  and  force  appear  first  and  must  be 
considered  as  essential  elements  in  physical  problems.  No  defini- 
tions of  these  familiar  elements  are  necessary  here. 

4.  Enumeration  of  the  Principles  and  Laws.     In  representing 
the  various  physical  magnitudes  by  numbers,  certain  agreements 
must  be  made  as  to  what  shall  be  considered  positive,  and  what 
negative.     The  axioms  of  ordinary  Geometry  will  be  considered 
as  being  true. 

The  fundamental  principles  upon  which  all  work  in  Theoretical 
Mechanics  may  be  made  to  depend  are  Newton's  three  Axioms,  or 
Laws  of  Motion.  The  first  two  laws  were  known  by  Galileo  and 
Huyghens,  although  they  were  for  the  first  time  announced 
together  in  all  their  completeness  by  Newton  in  the  Principia, 
in  1686.  These  laws  are  as  follows:* 

LAW  I.  Every  body  continues  in  its  state  of  rest,  or  of  uniform 
motion  in  a  straight  line,  unless  it  is  compelled  to  change  that  state  by 
a  force  impressed  upon  it. 

LAW  II.  The  rate  of  change  of  motion  is  proportional  to  the  force 
impressed,  and  takes  place  in  the  direction  of  the  straight  line  in  which 
the  force  acts. 

LAW  III.  To  every  action  there  is  an  equal  and  opposite  reaction; 
or,  the  mutual  actions  of  two  bodies  are  always  equal  and  oppositely 
directed. 

5.  Nature  of  the  Laws  of  Motion.     Newton  calls  the  Laws  of 
Motion  Axioms,  and  after  giving  each,  makes  a  few  remarks  con- 
cerning its  import.     Later  writers,  among  whom  are  Thomson  and 
Tait,f  regard  them  as  inferences  from  experience,  but  accept  New- 
ton's formulation  of  them  as  practically  final,  and  adopt  them 
in  the  precise  form  in  which  they  were  given  in  the  Principia.     A 
number  of  Continental  writers,  among  whom  is  Dr.  Ernest  Mach, 
have  given  profound  thought  to  the  fundamental  principles  of 

*  Other  fundamental  laws  may  be,  and  indeed  have  been,  employed;  but 
they  involve  more  difficult  mathematical  principles  at  the  very  start.  They 
are  such  as  d'Alembert's  principle,  Hamilton's  principle,  and  the  systems  of 
Kirchhoff,  Mach,  Hertz,  Boltzmann,  etc. 

t  Natural  Philosophy,  vol.  i.,  Art.  243. 


4:  REMARKS   ON    THE   FIRST  LAW   OF  MOTION.  [6 

Mechanics,  and  have  concluded  that  they  are  not  only  inductions  or 
simply  conventions,  but  that  Newton's  statement  of  them  is  some- 
what redundant,  and  lacks  scientific  directness  and  simplicity. 
There  is  no  suggestion,  however,  that  Newton's  Laws  of  Motion 
are  not  in  harmony  with  ordinary  astronomical  experience,  or  that 
they  cannot  be  made  the  basis  for  Celestial  Mechanics.  But  in 
some  branches  of  Physics,  particularly  in  Electricity  and  Light, 
certain  phenomena  are  not  fully  consistent  with  the  Newtonian 
principles,  and  they  have  recently  led  Einstein  and  others  to  the 
development  of  the  so-called  Principle  of  Relativity.  The  astro- 
nomical consequences  of  this  modification  of  the  principles  of 
Mechanics  are  very  slight  unless  the  time  under  consideration 
is  very  long,  and,  whether  they  are  true  or  not,  they  cannot  be 
considered  in  an  introduction  to  the  subject. 

6.  Remarks  on  the  First  Law  of  Motion.     In  the  first  law  the 
statement  that  a  body  subject  to  no  forces  moves  with  uniform 
motion,  may  be  regarded  as  a  definition  of  time.     For,  otherwise, 
it  is  implied  that  there  exists  some  method  of  measuring  time  in 
which  motion  is  not  involved.     Now  it  is  a  fact  that  in  all  the 
devices  actually  used  for  measuring  time  this  part  of  the  law  is  a 
fundamental  assumption.     For  example,  it  is  assumed  that  the 
earth  rotates  at  a  uniform  rate  because  there  is  no  force  acting 
upon  it  which  changes  the  rotation  sensibly.* 

The  second  part  of  the  law,  which  affirms  that  the  motion  is  in 
a  straight  line  when  the  body  is  subject  to  no  forces,  may  be  taken 
as  defining  a  straight  line,  if  it  is  assumed  that  it  is  possible  to 
determine  when  a  body  is  subject  to  no  forces;  or,  it  may  be  taken 
as  showing,  together  with  the  first  part,  whether  or  not  forces 
are  acting,  if  it  is  assumed  that  it  is  possible  to  give  an  independent 
definition  of  a  straight  line.  Either  alternative  leads  to  trouble- 
some difficulties  when  an  attempt  is  made  to  employ  strict  and 
consistent  definitions. 

7.  Remarks  on  the  Second  Law  of  Motion.     In  the  second  law 
the  statement  that  the  rate  of  change  of  motion  is  proportional  to 
the  force  impressed,  may  be  regarded  as  a  definition  of  the  relation 
between  force  and  matter  by  means  of  which  the  magnitude  of  a 
force,  or  the  amount  of  matter  in  a  body  can  be  measured,  accord- 
ing as  one  or  the  other  is  supposed  to  be  independently  known. 
By  rate  of  change  of  motion  is  meant  the  rate  of  change  of  velocity 

*  See  memoir  by  R.  S.  Woodward,  Astronomical  Journal,  vol.  xxi.  (1901). 


7]  REMARKS   ON   THE   SECOND   LAW   OF  MOTION.  5 

multiplied  by  the  mass  of  the  body  moved.  This  is  usually  called 
the  rate  of  change  of  momentum,  and  the  ideas  of  the  second  law 
may  be  expressed  by  saying,  the  rate  of  change  of  momentum  is 
proportional  to  the  force  impressed  and  takes  place  in  the  direction 
of  the  straight  line  in  which  the  force  acts.  Or,  the  acceleration  of 
motion  of  a  body  is  directly  proportional  to  the  force  to  which  it  is 
subject,  and  inversely  proportional  to  its  mass,  and  takes  place  in 
the  direction  in  which  the  force  acts. 

It  may  appear  at  first  thought  that  force  can  be  measured 
without  reference  to  velocity  generated,  and  it  is  true  in  a  sense. 
For  example,  the  force  with  which  gravity  draws  a  body  downward 
is  frequently  measured  by  the  stretching  of  a  coiled  spring,  or  the 
intensity  of  magnetic  action  can  be  measured  by  the  torsion  of  a 
fiber.  But  it  will  be  noticed  in  all  cases  of  this  kind  that  the 
law  of  reaction  of  the  machine  has  been  determined  in  some  other 
way.  This  may  not  have  been  directly  by  velocities  generated, 
but  it  ultimately  leads  back  to  it.  It  is  worthy  of  note  in  this 
connection  that  all  the  units  of  absolute  force,  as  the  dyne,  contain 
explicitly  in  their  definitions  the  idea  of  velocity  generated. 

In  the  statement  of  the  second  law  it  is  implied  that  the  effect 
of  a  force  is  exactly  the  same  in  whatever  condition  of  rest  or  of 
motion  the  body  may  be,  and  to  whatever  other  forces  it  may  be 
subject.  The  change  of  motion  of  a  body  acted  upon  by  a  number 
of  forces  is  the  same  at  the  end  of  an  interval  of  time  as  if  each 
force  acted  separately  for  the  same  time.  Hence  the  implication 
in  the  second  law  is,  if  any  number  of  forces  act  simultaneously  on 
a  body,  whether  it  is  at  rest  or  in  motion,  each  force  produces  the  same 
total  change  of  momentum  that  it  would  produce  if  it  alone  acted  on 
the  body  at  rest.  It  is  apparent  that  this  principle  leads  to  great 
simplifications  of  mechanical  problems,  for  in  accordance  with  it 
the  effects  of  the  various  forces  can  be  considered  separately. 

Newton  derived  the  parallelogram  of  forces  from  the  second 
law  of  motion.*  He  reasoned  that  as  forces  are  measured  by  the 
accelerations  which  they  produce,  the  resultant  of,  say,  two  forces 
should  be  measured  by  the  resultant  of  their  accelerations.  Since 
an  acceleration  has  magnitude  and  direction  it  can  be  represented 
by  a  directed  line,  or  vector.  The  resultant  of  two  forces  will 
then  be  represented  by  the  diagonal  of  a  parallelogram,  of  which 
two  adjacent  sides  represent  the  two  forces. 

*  Principia,  Cor.  i.  to  the  Laws  of  Motion. 


6  REMARKS   ON   THE   THIRD   LAW   OF  MOTION.  [8 

One  of  the  most  frequent  applications  of  the  parallelogram  of 
forces  is  in  the  subject  of  Statics,  which,  in  itself,  does  not  involve 
the  ideas  of  motion  and  time.  In  it  the  idea  of  mass  can  also  be 
entirely  eliminated.  Newton's  proof  of  the  parallelogram  of 
forces  has  been  objected  to  on  the  ground  that  it  requires  the 
introduction  of  the  fundamental  conceptions  of  a  much  more 
complicated  science  than  the  one  in  which  it  is  employed.  Among 
the  demonstrations  which  avoid  this  objectionable  feature  is  one 
due  to  Poisson,*  which  has  for  its  fundamental  assumption  the 
axiom  that  the  resultant  of  two  equal  forces  applied  at  a  point  is 
in  the  line  of  the  bisector  of  the  angle  which  they  make  with 
each  other.  Then  the  magnitude  of  the  resultant  is  derived,  and 
by  simple  processes  the  general  law  is  obtained. 

8.  Remarks  on  the  Third  Law  of  Motion.  The  first  two  of 
Newton's  laws  are  sufficient  for  the  determination  of  the  motion 
of  one  body  subject  to  any  number  of  known  forces;  but  another 
principle  is  needed  when  the  investigation  concerns  the  motion  of 
a  system  of  two  or  more  bodies  subject  to  their  mutual  interactions. 
The  third  law  of  motion  expresses  precisely  this  principle.  It  is 
that  if  one  body  presses  against  another  the  second  resists  the 
action  of  the  first  with  the  same  force.  And  also,  though  it  is 
not  so  easy  to  conceive  of  it,  if  one  body  acts  upon  another  through 
any  distance,  the  second  reacts  upon  the  first  with  an  equal  and 
oppositely  directed  force. 

Suppose  one  can  exert  a  given  force  at  will;  then,  by  the  second 
law  of  motion,  the  relative  masses  of  bodies  can  be  measured  since 
they  are  inversely  proportional  to  the  accelerations  which  equal 
forces  generate  in  them.  When  their  relative  masses  have  been 
found  the  third  law  can  be  tested  by  permitting  the  various  bodies 
to  act  upon  one  another  and  measuring  their  relative  accelera- 
tions. Newton  made  several  experiments  to  verify  the  law,  such 
as  measuring  the  rebounds  from  the  impacts  of  elastic  bodies,  and 
observing  the  accelerations  of  magnets  floating  in  basins  of  water. f 
The  chief  difficulty  in  the  experiments  arises  in  eliminating  forces 
external  to  the  system  under  consideration,  'and  evidently  they 
cannot  be  completely  removed.  Newton  also  concluded  from  a 
certain  course  of  reasoning  that  to  deny  the  third  law  would  be  to 
contradict  the  first. f 

Mach  points  out  that  there  is  no  accurate  means  of  measuring 
*  Traite  de  Mecanique,  vol.  I.,  p.  45  ei  seq. 
f  Principia,  Scholium  to  the  Laws  of  Motion. 


8]  REMARKS   ON   THE   THIRD   LAW   OF  MOTION.  7 

forces  except  by  the  accelerations  they  produce  in  masses,  and 
therefore  that  effectively  the  reasoning  in  the  preceding  paragraph 
is  in  a  circle.  He  objects  also  to  Newton's  definition  that  mass 
is  proportional  to  the  product  of  the  volume  and  the  density  of  a 
body.  He  prefers  to  rely  upon  experience  for  the  fact  that  two 
bodies  which  act  upon  each  other  produce  oppositely  directed 
accelerations,  and  to  define  the  relative  values  of  the  masses  as 
inversely  proportional  to  these  accelerations.  Experience  proves 
further  that  if  the  relative  masses  of  two  bodies  are  determined 
by  their  interactions  with  a  third,  the  ratio  is  the  same  whatever 
the  third  mass  may  be.  In  this  way,  when  one  body  is  taken  as 
the  unit  of  mass,  the  masses  of  all  other  bodies  can  be  uniquely 
determined.  These  views  have  much  to  commend  them. 

In  the  Scholium  appended  to  the  Laws  of  Motion  Newton  made 
some  remarks  concerning  an  important  feature  of  the  third  law. 
This  was  first  stated  in  a  manner  in  which  it  could  actually  be 
expressed  in  mathematical  symbols  by  d'Alembert  in  1742,  and 
has  ever  since  been  known  by  his  name.*  It  is  essentially  this: 
When  a  body  is  subject  to  an  acceleration,  it  may  be  regarded  as 
exerting  a  force  which  is  equal  and  opposite  to  the  force  by  which 
the  acceleration  is  produced.  This  may  be  considered  as  being 
true  whether  the  force  arises  from  another  body  forming  a  system 
with  the  one  under  consideration,  or  has  its  source  exterior  to  the 
system.  In  general,  in  a  system  of  any  number  of  bodies,  the 
resultants  of  all  the  applied  forces  are  equal  and  opposite  to  the 
reactions  of  the  respective  bodies.  In  other  words,  the  impressed 
forces  and  the  reactions,  or  the  expressed  forces,  form  systems 
which  are  in  equilibrium  for  each  body  and  for  the  whole  system. 
This  makes  the  whole  science  of  Dynamics,  in  form,  one  of  Statics, 
and  formulates  the  conditions  so  that  they  are  expressible  in 
mathematical  terms.  This  phrasing  of  the  third  law  of  motion 
has  been  made  the  starting  point  for  the  elegant  and  very  general 
investigations  of  Lagrange  in  the  subject  of  Dynamics.! 

The  primary  purpose  of  fundamental  principles  in  a  science  is  to 
coordinate  the  various  phenomena  by  stating  in  what  respects 
their  modes  of  occurring  are  common;  the  value  of  fundamental 
principles  depends  upon  the  completeness  of  the  coordination  of 
the  phenomena,  and  upon  the  readiness  with  which  they  lead  to 
the  discovery  of  unknown  facts;  the  characteristics  of  funda- 

*  See  Appell's  Mecanique,  vol.  n.,  chap.  xxm. 
f  Collected  Works,  vols.  xi.  and  xn. 


8  SPEED   AND   VELOCITY   IN    RECTILINEAR  MOTION.  [9 

mental  principles  should  be  that  they  are  self-consistent,  that 
they  are  consistent  with  every  observed  phenomenon,  and  that 
they  are  simple  and  not  redundant. 

Newton's  laws  coordinate  the  phenomena  of  the  mechanical 
sciences  in  a  remarkable  manner,  while  their  value  in  making 
discoveries  is  witnessed  by  the  brilliant  achievements  in  the 
physical  sciences  in  the  last  two  centuries  compared  to  the  slow 
and  uncertain  advances  of  all  the  ancients.  They  have  not  been 
found  to  be  mutually  contradictory,  and  they  are  consistent  with 
nearly  all  the  phenomena  which  have  been  so  far  observed;  they 
are  conspicuous  for  their  simplicity,  but  it  has  been  claimed  by 
some  that  they  are  in  certain  respects  redundant.  One  naturally 
wonders  whether  they  are  primary  and  fundamental  laws  of 
nature,  even  as  modified  by  the  principle  of  relativity.  In  view 
of  the  past  evolution  of  scientific  and  philosophical  ideas  one 
should  be  slow  in  affirming  that  any  statement  represents  ultimate 
and  absolute  truth.  The  fact  that  several  other  sets  of  funda- 
mental principles  have  been  made  the  bases  of  systems  of  me- 
chanics, points  to  the  possibility  that  perhaps  some  time  the 
Newtonian  system,  or  the  Newtonian  system  as  modified  by  the 
principle  of  relativity,  even  though  it  may  not  be  found  to  be  in 
error,  will  be  supplanted  by  a  simpler  one  even  in  elementary 
books. 

DEFINITIONS  AND  GENERAL  EQUATIONS. 

9.  Rectilinear  Motion,  Speed,  Velocity.  A  particle  is  in 
rectilinear  motion  when  it  always  lies  in  the  same  straight  line,  and 
when  its  distance  from  a  point  in  that  line  varies  with  the  time. 
It  moves  with  uniform  speed  if  it  passes  over  equal  distances  in 
equal  intervals  of  time,  whatever  their  length.  The  speed  is 
represented  by  a  positive  number,  and  is  measured  by  the  distance 
passed  over  in  a  unit  of  a  time.  The  velocity  of  a  particle  is  the 
directed  speed  with  which  it  moves,  and  is  positive  or  negative 
according  to  the  direction  of  the  motion.  Hence  in  uniform  motion 
the  velocity  is  given  by  the  equation 


Since  s  may  be  positive  or  negative,  v  may  be  positive  or  negative, 
and  the  speed  is  the  numerical  value  of  v.  The  same  value  of  v  is 
obtained  whatever  interval  of  time  is  taken  so  long  as  the  corre- 
sponding value  of  s  is  used. 


10]  ACCELERATION  IN  RECTILINEAR  MOTION.  9 

The  speed  and  velocity  are  variable  when  the  particle  does  not 
describe  equal  distances  in  equal  times;  and  it  is  necessary  to  define 
in  this  case  what  is  meant  by  the  speed  and  velocity  at  an  in- 
stant. Suppose  a  particle  passes  over  the  distance  As  in  the  time 
At,  and  suppose  the  interval  of  time  At  approaches  the  limit  zero  in 
such  a  manner  that  it  always  contains  the  instant  t.  Suppose, 
further,  that  for  every  At  the  corresponding  As  is  taken.  Then 
the  velocity  at  the  instant  t  is  defined  as 

/o\  r     / &s\       ds 

(2)  v  =  lim  [  -  -  )  =  -=- , 

A,=o  \At  J      dt' 

ds 
and  the  speed  is  the  numerical  value  of  -77 . 

Uniform  and  variable  velocity  may  be  defined  analytically  in 
the  following  manner.  The  distance  s,  counted  from  a  fixed  point, 
is  considered  as  a  function  of  the  time,  and  may  be  written 

s  =  0(0. 
Then  the  velocity  may  be  defined  by  the  equation 


Tt          ' 

where  <j>'(t)  is  the  derivative  of  0(0  with  respect  to  t.  The  velocity 
is  said  to  be  constant,  or  uniform,  if  0'(0  does  not  vary  with  t', 
or,  in  other  words,  if  0(0  involves  t  linearly  in  the  form  0(0  =«£+&, 
where  a  and  b  are  constants.  It  is  said  to  be  variable  if  the  value 
of  0'(0  changes  with  t. 

Some  agreement  must  be  made  to  denote  the  direction  of 
motion.  An  arbitrary  point  on  the  line  may  be  taken  as  the 
origin  and  the  distances  to  the  right  counted  as  positive,  and 
those  to  the  left,  negative.  With  this  convention,  if  the  value  of  s 
determining  the  position  of  the  body  increases  algebraically  with 
the  time  the  velocity  will  be  taken  positive;  if  the  value  of  s  de- 
creases as  the  time  increases  the  velocity  will  be  taken  negative. 
Then,  when  v  is  given  in  magnitude  and  sign,  the  speed  and  direc- 
tion of  motion  are  determined. 

10.  Acceleration  in  Rectilinear  Motion.  Acceleration  is  the 
rate  of  change  of  velocity,  and  may  be  constant  or  variable.  Since 
the  case  when  it  is  variable  includes  that  when  it  is  constant,  it 
will  be  sufficient  to  consider  the  former.  The  definition  of  acceler- 


10  SPEED   AND   VELOCITIES   IN   CURVILINEAR  MOTION.  [11 

ation  at  an  instant  t  is  similar  to  that  for  velocity,  and  is,  if  the 
acceleration  is  denoted  by  a, 


/o\ 
(3) 


r      f  Av\       dv 
a  =  hm  (  —  )  =  -=-  . 

A<=O  \  At  /       at 


By  means  of  (2)  and  (3)  it  follows  that 

d  /ds\       d2s 
a  =  dt(di)==dt*' 

There  must  be  an  agreement  regarding  the  sign  of  the  accelera- 
tion. When  the  velocity  increases  algebraically  as  the  time 
increases,  the  acceleration  will  be  taken  positive  ;  when  the  velocity 
decreases  algebraically  as  the  time  increases,  the  acceleration  will 
be  taken  negative. 

11.  Speed  and  Velocities  in  Curvilinear  Motion.  The  speed 
with  which  a  particle  moves  is  the  rate  at  which  it  describes  a 
curve.  If  v  represents  the  speed  in  this  case,  and  s  the  arc  of  the 
curve,  then 

ds 


<6>  -  -  dt 


where 


ds 


represents  the  numerical  value  of  -77 .      As  before,  the 


velocity  is  the  directed  speed  possessing  the  properties  of  vectors, 
and  may  be  represented  by  a  vector.*  The  vector  can  be  resolved 
uniquely  into  three  components  parallel  to  any  three  coordinate 
axes;  and  conversely,  the  three  components  can  be  compounded 
uniquely  into  the  vector.  In  other  words,  if  the  velocity  is  given, 
the  components  parallel  to  any  coordinate  axes  are  defined;. and 
the  components  parallel  to  any  non-coplanar  coordinate  axes  define 
the  velocity.  It  is  generally  simplest  to  use  rectangular  axes  and 
to  employ  the  components  of  velocity  parallel  to  them.  Let 
X,  M,  v  represent  the  angles  between  the  line  of  motion  and  the 
x,  y,  and  z-axes  respectively.  Then 

(ft  \  =  —      c  s     =  —      c  s  v  =  — 

Let  vx,  vy,  vz  represent  the  components  of  velocity  along  the  three 
axes.  That  is, 

*  Consult  Appell's  Mecanique,  vol.  I.,  p.  45  et  seq. 


12] 


ACCELERATION   IN   CURVILINEAR  MOTION. 


11 


(7) 


ds  dx      dx 

vx  =  v  cos  X  =  -77  -j-  =  -77  , 
dt  ds       dt 

ds  dy      dy 

Vy     =     V    COS  fJL     =    -TT   -J-    =    -77  , 


dz       dz 


From  these  equations  it  follows  that 


(8) 


There  must  be  an  agreement  as  to  a  positive  and  a  negative 
direction  along  each  of  the  three  coordinate  axes. 

12.  Acceleration  in  Curvilinear  Motion.  As  in  the  case  of 
velocities,  it  is  simplest  to  resolve  the  acceleration  into  component 
accelerations  parallel  to  the  coordinate  axes.  On  constructing  a 
notation  corresponding  to  that  used  in  Art.  11,  the  following 
equations  result: 

_  d2x  _  d2y  _  d2z 

OLx  ~  ,7/2  »          av  ~  dp  >          a*  ~  rffz  ' 

Hi  (Jili  (Jiil/ 


(8) 


The  numerical  value  of  the  whole  acceleration  is 


This  is  not,  in  general,  equal  to  the  component  of  acceleration 


d?s 


along  the  curve;  that  is,  to  -.     For,  from  (8)  it  follows  that 


ds 

V=dt 


whence,  by  differentiation, 

dx  d2x 
d2s  dt  dt2 


dz  d2z 


en) 


It2"     l/dx\2       idy\2       idz\ 
V\dt)        \dt  )        \dt) 

dx  d2x      dy  d2y      dz  d2z 


Thus,  when  the  components  of  acceleration  are  known,  the 
whole  acceleration  is  given  by  (10),  and  the  acceleration  along 


12 


POLAR  COMPONENTS  OF  VELOCITY 


[13 


the  curve  by  (11).  The  fact  that  the  two  are  different,  in  general, 
may  cause  some  surprise  at  first  thought.  But  the  matter  becomes 
clear  if  a  body  moving  in  a  circle  with  constant  speed  is  con- 
sidered. The  acceleration  along  the  curve  is  zero  because  the 
speed  is  supposed  not  to  change;  but  the  acceleration  is  not  zero 
because  the  body  does  not  move  in  a  straight  line. 

13.  The  Components  of  Velocity  Along  and  Perpendicular  to 
the  Radius  Vector.  Suppose  the  path  of  the  particle  is  in  the 
iC2/-plane,  and  let  the  polar  coordinates  be  r  and  6.  Then 

(12)  x  =  r  cos  8,    y  =  r  sin  6. 

The  components  of  velocity  are  therefore 

dx  .     „  dd  .          „  dr 

dt 
(13) 


-TT  =  v, 


•    Qdd 

—  r  sin  6  -TT 


Let  QP  be  an  arc  of  the  curve  described  by  the  moving  particle. 
When  the  particle  is  at  P,  it  is  moving  in  the  direction  PV,  and 
the  velocity  may  be  represented  by  the  vector  PV.  Let  vr  and  v& 


Fig.  1. 

represent  the  components  of  velocity  along  and  perpendicular  to 
the  radius  vector.     The  resultant  of  the  vectors  vr  and  v0  is  equal 

(i'lr  r/?y 

to  the  resultant  of  the  vectors  3-  and  ~  ,  that  is,  to  PV.     The 

at  at 

sum  of  the  projections  of  vr  and  VQ  upon  any  line  equals  the  sum 

of  the  projections  of  3-  and  -^  upon  the  same  line.     Therefore, 
at  at 

projecting  vr  and  v&  upon  the  x  and  y-axes,  it  follows  that 


(14) 


-j7  =  vr  cos  6  —  v^  sin  6, 
dt 

dii 

-r  =  vr  sin  0  +  VQ  cos  6. 

at 


14]  POLAR   COMPONENTS   OF  ACCELERATION.  13 

On  comparing  (13)  and  (14),  the  required  components  of  velocity 

are  found  to  be 

dr 


The  square  of  the  speed  is 


The  components  of  velocity,  vr  and  vd,  can  be  found  in  terms 
of  the  components  parallel  to  the  x  and  ?/-axes  by  multiplying 
equations  (14)  by  cos  6  and  sin  6  respectively  and  adding,  and 
then  by  —  sin  d  and  cos  d  and  adding.  The  results  are 

.  dx  .     .     n  di/ 
vr  =  +  cos  6  -=7  -f-  sm  6  —-, 
at  at 

(16) 

.     _  dx  .          ndy 
ve  =  —  sin  6  -=-  -f  cos  6  -37. 
at  at 

14.  The  Components  of  Acceleration.  The  derivatives  of 
equations  (13)  are 


(Px      Y(Pr        /<«  VI  [  d*8  ,ndrd8l   . 

a*  =  Hi?  =  [df-  r  (di)  \cose-  [rdt*+2  Jt  Jt\  sm  "• 

<Pr        /d»\21   . 

de  -  T  (jt)  J  sm  "• 


'•        *v_\_*9,n*d» 


Let  ctr  and  a&  represent  the  components  of  acceleration  along 
and  perpendicular  to  the  radius  vector.  As  in  Art.  13,  it  follows 
from  the  composition  and  resolution  of  vectors  that 

{ax  =  OLT  cos  d  —  O.Q  sin  6, 
ay  =  ar  sin  6  +  ae  cos  6. 
On  comparing  (17)  and  (18),  it  is  found  that 


The  components  of  acceleration  along  and  perpendicular  to  the 


14 


PARTICLE   MOVING   IN  A   CIRCLE. 


[15 


radius  vector  in  terms  of  the  components  parallel  to  the  x  and 
2/-axes  are  found  from  (17)  to  be 


(20) 


ar= 


«=  - 


— 


By  similar  processes  the  components  of  velocity  and  acceleration 
parallel  to  any  lines  can  be  found. 

15.  Application  to  a  Particle  Moving  in  a  Circle  with  Uniform 
Speed.  Suppose  the  particle  moves  with  uniform  speed  in  a  circle 
around  the  origin  as  center;  it  is  required  to  determine  the  com- 


axis 


Fig.  2. 

ponents  of  velocity  and  acceleration  parallel  to  the  x  and  y-axes, 
and  parallel  and  perpendicular  to  the  radius.  Let  R  represent 
the  radius  of  the  circle;  then 

x  =  R  cos  0,        y  =  R  sin  6. 

Since  the  speed  is  uniform  the  angle  6  is  proportional  to  the  time, 
or  6  =  ct.  The  coordinates  become 

(21)  x  =  R  cos  (ct),         y  =  R  sin  (ct). 

3  n  /jj? 

Since  -r  =  c  and  -r-  =  0,  the  components  of  velocity  parallel  to 

CtL  (juL 

the  x  and  ?/-axes  are  found  from  (13)  to  be 

(22)  vx  =  -  Re  sin  (ct),         vv  =  Re  cos  (ct). 
From  (15)  it  is  found  that 

(23)  vr  =  0,         ve  =  Re. 

The  components  of  acceleration  parallel  to  the  x  and  i/-axes, 
which  are  given  by  (17),  are 


16] 


AREAL   VELOCITY. 


15 


(24) 


ax  —  —  Rc2  cos  (ct), 
a.y  —  —  Re2  sin  (ct). 


And  from  (19)  it  is  found  that 


(25) 


=  —  Rc2, 


=  0. 


It  will  be  observed  that,  although  the  speed  is  uniform  in  this 
case,  the  velocity  with  respect  to  fixed  axes  is  not  constant,  and 
the  acceleration  is  not  zero.  If  it  is  assumed  that  an  exterior 
force  is  the  only  cause  of  the  change  of  motion,  or  of  acceleration 
of  a  particle,  then  it  follows  that  a  particle  cannot  move  in  a 
circle  with  uniform  speed  unless  it  is  subject  to  some  force.  It 
follows  also  from  (25)  and  the  second  law  of  motion  that  the  force 
continually  acts  in  a  line  which  passes  through  the  center  of  the 
circle. 

16.  The  Areal  Velocity.  The  rate  at  which  the  radius  vector 
from  a  fixed  point  to  the  moving  particle  describes  a  surface  is 


y  -  oris 


X-axis 


Fig.  3. 

called  the  areal  velocity  with  respect  to  the  point.  Suppose  the 
particle  moves  in  the  :n/-plane.  Let  AA  represent  the  area  of  the 
triangle  OPQ  swept  over  by  the  radius  vector  in  the  interval  of 
time  A*.  Then 


•sin(A0); 


whence 
(26) 


r    sin  (A0)    A0 


A*  2  A0         Ar 

As  the  angle  A0  diminishes  the  ratio  of  the  area  of  the  triangle  to 


16  MOTION   OF  A   PARTICLE   IN   AN   ELLIPSE.  [17 

that  of  the  sector  approaches  unity  as  a  limit.     The  limit  of 
r'  is  r,  and  the  limit  of  -  is  unity.     Equation  (26)  gives,  on 

passing  to  the  limit  AZ  =  0  in  both  members, 

(27)  —  =  -r 2  — 

dt       2T  dt 

as  the  expression  for  the  areal  velocity.     On  changing  to  rect- 
angular coordinates  by  the  substitution 

r  =  Vz2  +  2/2,        tan  0  =  -  , 

£6 

equation  (27)  becomes 

(28)  ^  =  l/xg_^ 

If  the  motion  is  not  in  the  zt/-plane  the  projections  of  the  areal 
velocity  upon  the  three  fundamental  planes  are  used.  They  are 
respectively 

dA  xy  _  1  /    dy  _      dx\ 

~dT  ~2\Xdt~~ydt)' 

dAyz      1  /    dz        dy 


(29) 


dt 

dAzx 
dt 


_  1  /    dx  _     dz\ 
~2\Zdt       Xdt)' 


In  certain  mechanical  problems  the  body  considered  moves  so 
that  the  areal  velocity  is  constant  if  the  origjn  is  properly  chosen. 
In  this  case  it  is  said  that  the  body  obeys  the  law  of  areas  with 
respect  to  the  origin.  That  is, 

r2  -7-  =  constant. 
at 

It  follows  from  this  equation  and  (19)  that  in  this  case 

ae  =  0; 
that  is,  the  acceleration  perpendicular  to  the  radius  vector  is  zero. 

17.  Application  to  Motion  in  an  Ellipse.  Suppose  a  particle 
moves  in  an  ellipse  whose  semi-axes  are  a  and  b  in  such  a  manner 
that  it  obeys  the  law  of  areas  with  respect  to  the  center  of  the 
ellipse  as  origin;  it  is  required  to  find  the  components  of  accelera- 


PROBLEMS.  17 

tion  along  and  perpendicular  to  the  radius  vector.     The  equation 
of  the  ellipse  may  be  written  in  the  parametric  form 

(30)  x  =  a  cos  0,        y  =  b  sin  0; 

for,  if  0  is  eliminated,  the  ordinary  equation 


is  found.     It  follows  from  (30)  that 

/0-i\  dx  .        d<f>          dy 

(31)  _=-aSm0^,        J- 

On  substituting  (30)  and  (31)  in  the  expression  for  the  law  of  areas, 


^   it  is  found  that 


d$  =  c_ 
dt  ~  ab 


The  integral  of  this  equation  is 


+  cz; 


and  if  <£  =  0  when  t  =  0,  then  Cz  =  0  and  <£  =  c\t. 

On  substituting  the  final  expression  for  <£  in  (30),  it  is  found  that 
'd2x 


-  Ci'a  cos  0  =  - 

—  Ci26  sin  cj)  =  —  c^y. 

If  these  values  of  the  derivatives  are  substituted  in  (20)  the 
components  of  acceleration  along  and  perpendicular  to  the  radius 
vector  are  found  to  be 

r  ar  =  —  Ci2r, 

Oi&    =    0. 

I.     PROBLEMS. 

1.  A  particle  moves  with  uniform  speed  along  a  helix  traced  on  a  circular 
cylinder  whose  radius  is  R;  find  the  components  of  velocity  and  acceleration 
parallel  to  the  x,  y,  and  z-axes.  The  equations  of  the  helix  are 

x  =  R  cos  a),     y  =  R  sin  o>,     z  =  hu>. 
3 


18 


PROBLEMS. 


[  vx  —  — 

Ans.     •{ 

I  ax  =  - 


—  Re  sin  (d),          vy  =  -\-  Re  cos  (d),         vz  =  he; 
Re2  cos  (d),        ay  =  —  Re2  sin  (d),        az  =  0. 


2.  A  particle  moves  in  the  ellipse  whose  parameter  and  eccentricity  are 
p  and  e  with  uniform  angular  speed  with  respect  to  one  of  the  foci  as  origin; 
find  the  components  of  velocity  and  acceleration  along  and  perpendicular 
to  the  radius  vector  and  parallel  to  the  x  and  y-axes  in  terms  of  the  radius 
vector  and  the  time. 


Ans. 


v6  =  re; 


vx  =  -  cr  sin  (d)  +  —  •  r2  sin  (2d), 

€>C 

vy  =  cr  cos  (d)  H  —  •  r2  sin2  (d)  ; 


2ec2 


(d)  +     f-  r3  sin2  (d)  -  c2r, 


»*sin  (ct); 


otx=  -  c*r  cos  (d)  +  —  •  r2  -  --  •  r2  sin2  (d) 

2e2c2 
H  --  g-  •  r3  sin2  (d)  cos  (d), 


sn 


sn3 


3.  A  'particle  moves  in  an  ellipse  in  such  a  manner  that  it  obeys  the  law 
of  areas  with  respect  to  one  of  the  foci  as  an  origin;  it  is  required  to  find  the 
components  of  velocity  and  acceleratio  i  along  and  perpendicular  to  the  radius 
vector  and  parallel  to  the  axes  in  terms  of  the  coordinates. 


Ans. 


eA    . 

vr  =  —  sin  d, 


> 


eA   .    nn      A  sin  6  eA   .  ,      .  A  cos  6 

v,  =  ^  sin  20 j^,        y,  =  -  sm2  0  +  — —  ; 


»•  - 


A'     1 

'?' 


=0; 


«i  = 


A2     cos  0 


p         r2 


A2     sing 
p    '     r2 


4.  The  accelerations  along  the  x  and  y-axes  are  the  derivatives  of  the 
velocities  along  these  axes;  why  are  not  the  accelerations  along  and  per- 
pendicular to  the  radius  vector  given  by  the  derivatives  of  the  velocities  in 
these  respective  directions?  Find  the  accelerations  along  axes  rotating  with 
the  angular  velocity  unity  in  terms  of  the  accelerations  with  respect  to  fixed 
axes. 


18] 


CENTER   OF   MASS   OF   SYSTEMS   OF   PARTICLES. 


19 


18.  Center  of  Mass  of  n  Equal  Particles.  The  center  of  mass 
of  a  system  of  equal  particles  will  be  defined  as  that  point  whose 
distance  from  any  plane  is  equal  to  the  average  distance  of  all 
of  the  particles  from  that  plane.  This  must  be  true  then  for  the 
three  reference  planes.  Let  (xi,  yi,  Zi),  (xz,  yz,  Zz),  etc.,  represent 
the  rectangular  coordinates  of  the  various  particles,  and  x,  y,  z 
the  rectangular  coordinates  of  their  center  of  mass;  then  by  the 
definition 


(32) 


X  = 


y  = 


+  X2 


+  X, 


—   


Zi 


Z2 


i-l 

n 


Suppose  the  mass  of  each  particle  is  m,  and  let  M  represent  the 
mass  of  the  whole  system,  or  M  =  nm.  On  multiplying  the 
numerators  and  denominators  by  m,  equations  (32)  become 


(33) 


m 


x  = 


y  = 


z  = 


nm 

n 


M 


nm 


M 


nm 


=1_ 
M 


It  remains  to  show  that  the  distance  from  the  point  (x,  y,  z) 
to  any  other  plane  is  also  the  average  distance  of  the  particles 
from  the  plane.  The  equation  of  any  other  plane  is 


ax 


by  +  cz  +  d  =  0. 
The  distance  of  the  point  (x,  y,  z)  from  this  plane  is 


(34) 


-  _  ax  +  by  +  cz  +  d 
Va2  +  62  +  c2 


20 


CENTER   OF  MASS   OF   SYSTEMS   OF   PARTICLES. 


[19 


and  similarly,  the  distance  of  the  point  (Xi,  yi}  z»)  from  the  same 
plane  is 

,*n  j       aXi  +  ^  +  CZi  +  d 

(35)  di  =  —  .  2  2 

It  follows  from  equations  (32),  (34) ;  and  (35)  that 


+  62  +  c2 


n 


Therefore  the  point  (x,  y,  z)  denned  by  (32)  satisfies  the  definition 
of  center  of  mass  with  respect  to  all  planes. 

19.  Center  of  Mass  of  Unequal  Particles.  There  are  two 
cases,  (a)  that  in  which  the  masses  are  commensurable,  and  (fc) 
that  in  which  the  masses  are  incommensurable. 

(a)  Select  a  unit  m  in  terms  of  which  all  the  n  masses  can  be 
expressed  integrally.  Suppose  the  first  mass  is  p\m,  the  second 
p2m,  etc.,  and  let  pirn  =  mi,  p2m  =  m2,  etc.  The  system  may  be 
thought  of  as  made  up  of  p\  +  p^  +  •  •  •  particles  each  of  mass  m, 
and  consequently,  by  Art.  18, 


(36) 


M 


M      ' 


%    — 


M 


(b)  Select  an  arbitrary  unit  m  smaller  than  any  one  of  the 
n  masses.  They  will  be  expressible  in  terms  of  it  plus  certain 
remainders.  If  the  remainders  are  neglected  equations  (36)  give 
the  center  of  mass.  Take  as  a  new  unit  any  submultiple  of  m 
and  the  remainders  will  remain  the  same,  or  be  diminished, 
depending  on  their  magnitudes.  The  submultiple  of  m  can  be 
taken  so  small  that  every  remainder  is  smaller  than  any  assigned 


19]  CENTER  OF  MASS   OF  SYSTEMS  OF   PARTICLES.  21 

quantity.  Equations  (36)  continually  hold  where  the  m;  are  the 
masses  of  the  bodies  minus  the  remainders.  As  the  submultiples 
of  m  approach  zero  as  a  limit,  the  sum  of  the  remainders  approaches 
zero  as  a  limit,  and  the  expressions  (36)  approach  as  limits  the 
expressions  in  which  the  mt-  are  the  actual  masses  of  the  particles. 
Therefore  in  all  cases  equations  (36)  give  a  point  which  satisfies 
the  definition  of  center  of  mass. 

The  fact  that  if  the  definition  of  center  of  mass  is  fulfilled  for 
the  three  reference  planes,  it  is  also  fulfilled  for  every  other  plane 
can  easily  be  proved  without  recourse  to  the  general  formula  for 
the  distance  from  any  point  to  any  plane.  It  is  to  be  observed 
that  the  i/z-plane,  for  example,  may  be  brought  into  any  position 
whatever  by  a  change  of  origin  and  a  succession  of  rotations  of 
the  coordinate  system  around  the  various  axes.  It  will  be  neces- 
sary to  show,  then,  that  equations  (36)  are  not  changed  in  form 
(1)  by  a  change  of  origin,  and  (2)  by  a  rotation  around  one  of  the 
axes. 

(1)  Transfer  the  origin  along  the  #-axis  through  the  distance  a. 
The  substitution  which  accomplishes  the  transfer  is  x  =  x'  +  a, 
and  the  first  equation  of  (36)  becomes 

n 

Sm,i(xi  +  a) 

~    _+_   n    - 

whence 


~    _+_   n    -   _  -  __  L          =  . 

M  MM' 


n 


M 

which  has  the  same  form  as  before. 

(2)  Rotate  trie  x  and  2/-axes  around  the  z-axis  through  the 
angle  6.     The  substitution  which  accomplishes  the  rotation  is 

{x  =  x'  cos  9  —  yf  sin  0, 
y  =  x'  sin  6  -\-  y'  cos  6. 
The  first  two  equations  of  (36)  become  by  this  transformation 


x'  cos  6  —  y'  sin  0  =  cos  6  t=      --  sin  0 


x'  sin  e  +  y'  cos  B  =  sin  0  =_ (_  cos  6 


M  M 


22 


CENTER  OF   GRAVITY. 


[20 


On  solving  these  equations  it  is  found  that 


y'  = 


M 


M 


Therefore  the  point  (x,  y,  z)  satisfies  the  definition  of  center  of 
mass  with  respect  to  every  plane. 

20.  The  Center  of  Gravity.  The  members  of  a  system  of 
particles  which  are  near  together  at  the  surface  of  the  earth  are 
subject  to  forces  downward  which  are  sensibly  parallel  and  pro- 
portional to  their  respective  masses.  The  weight,  or  gravity,  of  a 
particle  will  be  defined  as  the  intensity  of  the  vertical  force  /, 


Fig.  4. 

which  is  the  product  of  the  mass  m  of  the  particle  and  its  accelera- 
tion g.  The  center  of  gravity  of  the  system  will  be  defined  as  the 
point  such  that,  if  the  members  of  the  system  were  rigidly  con- 
nected and  the  sum  of  all  the  forces  were  applied  at  this  point, 
then  the  effect  on  the  motion  of  the  system  would  be  the  same  as 
that  of  the  original  forces  for  all  orientations  of  the  system. 

It  will  now  be  shown  that  the  center  of  gravity  coincides  with 
the  center  of  mass.  Consider  two  parallel  forces  /i  and  /2  acting 
upon  the  rigid  system  M  at  the  points  Pi  and  P2.  Resolve  these 
two  forces  into  the  components  /  and  g\,  and  /  and  02  respectively. 
The  components  /,  being  equal  and  opposite,  destroy  each  other. 
Then  the  components  g\  and  #2  may  be  regarded  as  acting  at  A. 
Resolve  them  again  so  that  the  oppositely  directed  components 


20] 


CENTER   OF   GRAVITY. 


23 


shall  be  equal  and  lie  in  a  line  parallel  to  PiP2;  then  the  other  com- 
ponents will  lie  in  the  same  line  AB,  which  is  parallel  to  the 
direction  of  the  original  forces  /i  and  /2,  and  will  be  equal  respec- 
tively to  /i  and  /2.  Therefore  the  resultant  of  /i  and  /2  is  equal  to 
/!  +  /2  in  magnitude  and  direction.  It  is  found  from  similar 

triangles  that 

fi=AB_         f± 

7~PiB>        f 

whence,  by  division, 


The  solution  for  x  gives 


If  the  resultant  of  these  two  forces  be  united  with  a  third  force  /3, 
•the  point  where  their  sum  may  be  applied  with  the  same  effects  is 
found  in  a  similar  manner  to  be  given  by 


/1+/2+/3 

and  so  on  for  any  number  of  forces.     Similar  equations  are  true 
for  parallel  forces  acting  in  any  other  direction. 

Suppose  there  are  n  particles  m»  subject  to  n  parallel  forces  /,- 
due  to  the  attraction  of  the  earth.  The  coordinates  of  their 
center  of  gravity  with  respect  to  the  origin  are  given  by 


(37) 


M 


y  = 


M 


M 


The  center  of  gravity  is  thus  seen  to  be  coincident  with  the  center 
of  mass;  nevertheless  this  would  not  in  general  be  true  if  the  sys- 
tem were  not  in  such  a  position  that  the  accelerations  to  which 


24  CENTER  OF  MASS  OF  A   CONTINUOUS  BODY.  [21 

its  various  members  are  subject  were  both  equal  and  parallel. 
Euler  (1707-1783)  proposed  the  designation  of  center  of  inertia  for 
the  center  of  mass. 

21.  Center  of  Mass  of  a  Continuous  Body.  As  the  particles 
of  a  system  become  more  and  more  numerous  and  nearer  together 
it  approaches  as  a  limit  a  continuous  body.  In  the  case  of  the 
ordinary  bodies  of  mechanics  the  particles  are  innumerable  and 
indistinguishably  close  together;  on  this  account  such  bodies  are 
treated  as  continuous  masses.  For  continuous  masses,  therefore, 
the  limits  of  expressions  (37),  as  nii  approaches  zero,  must  be 
taken.  At  the  limit  m  becomes  dm  and  the  sum  becomes  the  defi- 
nite integral.  The  equations  which  give  the  center  of  mass  are 
therefore 


(38) 


_  =  fxdm 

fdm  : 

_  Jydm 
~  fdm  '• 


= 
' 


fdm  ' 

where  the  integrals  are  to  be  extended  throughout  the  whole  body. 
When  the  body  is  homogeneous  the  density  is  the  quotient  of 
any  portion  of  the  mass  divided  by  its  volume.  When  the  body 
is  not  homogeneous  the  mean  density  is  the  quotient  of  the  whole 
mass  divided  by  the  whole  volume.  The  density  at  any  point  is 
the  limit  of  the  mean  density  of  a  volume  including  the  point  in 
question  when  this  volume  approaches  zero  as  a  limit.  If  the 
density  is  represented  by  cr,  the  element  of  mass  is,  when  expressed 
in  rectangular  coordinates, 

dm  =  (rdxdydz. 
Then  equations  (38)  become 

fffffx  dx  dy  dz 


(39) 


=  fffvdxdydz  ' 

=  fffvydxdydz 
~  fffcrdxdydz  ' 

-  = 


fffadxdydz 

The  limits  of  the  integrals  depend  upon  the  shape  of  the  body, 
and  a  must  be  expressed  as  a  function  of  the  coordinates. 


21] 


CENTER  OF  MASS  OF  A  CONTINUOUS  BODY. 


25 


In  certain  problems  the  integrations  are  performed  more  simply 
if  polar  coordinates  are  employed.  The  element  of  mass  when 
expressed  in  polar  coordinates  is 

dm  =  a  -  ab  •  be  -  cd. 


Fig.  5. 
It  is  seen  from  the  figure  that 

ab  =  dr, 
be  =  rd(f>, 


cd  =  r  cos  <f>dd. 
dm  —  or2  cos  0  d<j>  d&  dr, 


Therefore 

(40) 
and 

Ix  =  r  cos  0  cos  6j 
y  =  r  cos  <f>  sin  0, 
z  =  r  sin  <f>. 

Therefore  equations  (38)  become 

///or3  cos2  <fr  cos  6  d$  dB  dr 
///or2  cos  0  d<j>  dd  dr 

fffvr3  cos2  </>  sin  0  d<f>  d0  dr 


(42) 


y 


fffo-r2  cos  0  d(f>  dd  dr 


.  _  ///or3  sin  4>  cos  <ft  d<f>  dB  dr 
fffar2  cos  <f>d<t>d8  dr 


26 


PLANES  AND  AXES  OF   SYMMETRY. 


[22 


The  density  a  must  be  expressed  as  a  function  of  the  coordinates, 
and  the  limits  must  be  so  taken  that  the  whole  body  is  included. 
If  the  body  is  a  line  or  a  surface  the  equations  admit  of  important 
simplifications. 

22.  Planes  and  Axes  of  Symmetry.     If  a  homogeneous  body 
is  symmetrical  with  respect  to  any  plane,  the  center  of  mass  is  in 
that  plane,  because  each  element  of  mass  on  one  side  of  the  plane 
can  be  paired  with  a  corresponding  element  of  mass  on  the  other 
side,  and  the  whole  body  can  be  divided  up  into  such  paired  ele- 
ments.    This  plane  is  called  a  plane  of  symmetry.     If  a  homo- 
geneous body  is  symmetrical  with  respect  to  two  planes,  the  center 
of  mass  is  in  the  line  of  their  intersection.     This  line  is  called  an 
axis  of  symmetry.     If  a  homogeneous  body  is  symmetrical  with 
respect  to  three  planes,  intersecting  in  a  point,  the  center  of  mass 
is  at  their  point  of  intersection.     From  the  consideration  of  the 
planes  and  axes  of  symmetry  the  centers  of  mass  of  many  of  the 
simple  figures  can  be  inferred  without  employing  the  methods  of 
integration. 

23.  Application  to  a  Non-Homogeneous  Cube.    Suppose  the 
density  varies  directly  as  the  square  of  the  distance  from  one  of  the 
faces  of  the  cube.     Take  the  origin  at  one  of  the  corners  and  let 
the  i/2-plane  be  the  face  of  zero  density.     Then  a  =  kx2,  where 
k  is  the  density  at  unit  distance.     Suppose  the  edge  of  the  cube 
equals  a;  then  equations  (39)  become 


x  = 


k  I    J     I    x3  dx  dy  dz 


x2y  dx  dy  dz 


x2  dx  dy  dz 


z  — 


- 

These  equations  become,  after  integrating  and  substituting  the 
limits, 


_  _a 
Z~2' 


24] 


APPLICATION   TO   THE   OCTANT   OF   A   SPHERE. 


27 


If  polar  coordinates  were  used  in  this  problem  the  upper  limits 
of  the  integrals  would  be  much  more  complicated  than  they  are 
with  rectangular  coordinates,  and  the  integration  would  be 
correspondingly  more  difficult. 

24.  Application  to  the  Octant  of  a  Sphere.  Suppose  the  sphere 
is  homogeneous  and  that  the  density  equals  unity.  It  is  preferable 
to  use  polar  coordinates  in  this  example,  although  it  is  by  no 
means  necessary.  Either  (39)  or  (42)  can  be  used  in  any  problem, 
and  the  choice  should  be  determined  by  the  form  that  the  limits 
take  in  the  two  cases.  It  is  desirable  to  have  them  all  constants 
when  they  can  be  made  so.  If  the  origin  is  taken  at  the  center 
of  the  sphere,  and  if  the  radius  is  a,  equations  (42)  become 


x  = 


IT  IT 

Jo    Jo    Jo 


r3  cos2^  cos6d<f>dddr 


y  = 


z  = 


m 

Jo    Jo    Jo 

ITf 

Jo    Jo    Jo 


r2  cos  4>  d(f>  d6  dr 


r3  cos2  0  sin  6  d(f>  dd  dr 


r2  cos  <f>  d<j>  dd  dr 


r3  sin  0  cos  <£  d(j>  dd  dr 


7T  7T 

m 


r2  cos  <£  d(l>  dd  dr 


Since  the  mass  of  a  homogeneous  sphere  with  radius  a  and  density 
unity  is  fira3,  each  of  the  denominators  of  these  expressions  equals 
^wa3.  This  can  be  verified  at  once  by  integration.  On  integrating 
the  numerators  with  respect  to  $  and  substituting  the  limits,  the 
equations  become 


r3  sin  6  d6  dr 


IT     , 

6° 


7T      , 

r 


28  PROBLEMS. 

On  integrating  with  respect  to  6,  these  equations  give 

7T      /*«  7T      /"«     ,  7T      /"« 

^ar  r^dr 

^  =  4  Jo  .  =4  J»  g  =4  Jo 

T      3  7T      ,  7T      3 

6  6  6 

and,  finally,  the  integration  with  respect  to  r  gives 
*  =  y  =  *  =  fa- 

The  octant  of  a  sphere  has  three  planes  of  symmetry,  viz.,  the 
planes  defined  by  the  center  of  the  sphere,  the  vertices  of  the 
bounding  spherical  triangle,  and  the  centers  of  their  respective 
opposite  sides.  Since  these  three  planes  intersect  not  only  in  a 
point  but  also  in  a  line,  they  do  not  fully  determine  the  center  of 
mass. 

As  nearly  all  the  masses  occurring  in  astronomical  problems  are 
spheres  or  oblate  spheroids  with  three  planes  of  symmetry  which 
intersect  in  a  point  but  not  in  a  line,  the  applications  of  the  for- 
mulas just  given  are  extremely  simple,  and  no  more  examples 
need  be  solved. 


H.    PROBLEMS. 

1.  Find  the  center  of  mass  of  a  fine  straight  wire  of  length  R  whose  density 
varies  directly  as  the  nth  power  of  the  distance  from  one  end. 

Ans.     — -.— _  R  from  the  end  of  zero  density. 
n  +  2 

2.  Find  the  coordinates  of  the  center  of  mass  of  a  fine  wire  of  uniform 
density  bent  into  a  quadrant  of  a  circle  of  radius  R. 

2R 

Ans.     x  =  y  =  —  f 

where  the  origin  is  at  the  center  of  the  circle. 

3.  Find  the  coordinates  of  the  center  of  mass  of  a  thin  plate  of  uniform 
density,  having  the  form  of  a  quadrant  of  an  ellipse  whose  semi-axes  are 
a  and  6. 

f_      4a 
5-§r' 
Ans.     1 

-       46 


25]  HISTORICAL  SKETCH   TO   NEWTON.  29 

4.  Find  the  coordinates  of  the  center  of  mass  of  a  thin  plate  of  uniform 
density,  having  the  form  of  a  complete  loop  of  the  lemniscate  whose  equation 
is  r2  =  a2  cos  20. 

Ans.     J  2* ' 

I  ?/  =  0. 

5.  Find  the  coordinates  of  the  center  of  mass  of  an  octant  of  an  ellipsoid 
of  uniform  density  whose  semi-axes  are  a,  b,  c. 

f  -       3a 

x  =  — . 


Ans. 


_       36 


3c 

Z   =  TT  • 


6.  Find  the  coordinates  of  the  center  of  mass  of  an  octant  of  a  sphere  of 
radius  R  whose  density  varies  directly  as  the  nth  power  of  the  distance  from 
the  center. 

_       _       _      n +3     R 
Ans.     *-y  _«__.-. 

7.  Find  the  coordinates  of  the  center  of  mass  of  a  paraboloid  of  revolution 
cut  off  by  a  plane  perpendicular  to  its  axis. 


(x  =  \h, 
Ans.     •<  _       _  ' 

(y  =  z  =  0, 


where  h  is  the  distance  from  the  vertex  of  the  paraboloid  to  the  plane. 

8.  Find  the  coordinates  of  the  center  of  mass  of  a  right  circular  cone  whose 
height  is  h  and  whose  radius  is  R. 

9.  Find  the  coordinates  of  the  center  of  mass  of  a  double  convex  lens  of 
homogeneous  glass  whose  surfaces  are  spheres  having  the  radii  ri  and  r2  =  2rt 

and  whose  thickness  at  the  center  is  -^—. — - . 

4 

10.  In  a  concave-convex  lens  the  radius  of  curvature  of  the  convex  and  con- 
cave surfaces  are  n  and  r2  >  n.     Determine  the  thickness  and  diameter  of  the 
lens  so  that  the  center  of  mass  shall  be  in  the  concave  surface. 

HISTORICAL  SKETCH  FROM  ANCIENT  TIMES  TO  NEWTON. 
25.  The  Two  Divisions  of  the  History.  The  history  of  the 
development  of  Celestial  Mechanics  is  naturally  divided  into  two 
distinct  parts.  The  one  is  concerned  with  the  progress  of  knowl- 
edge about  the  purely  formal  aspect  of  the  universe,  the  natural 
divisions  of  time,  the  configurations  of  the  constellations,  and 
the  determination  of  the  paths  and  periods  of  the  planets  in  their 


30  HISTORICAL   SKETCH.  [26 

motions;  the  other  treats  of  the  efforts  at,  and  the  success  in,  attain- 
ing correct  ideas  regarding  the  physical  aspects  of  natural  phe- 
nomena, the  fundamental  properties  of  force,  matter,  space,  and 
time,  and,  in  particular,  the  way  in  which  they  are  related.  It  is 
true  that  these  two  lines  in  the  development  of  astrcnomical 
science  have  not  always  been  kept  distinct  by  those  who  have 
cultivated  them;  on  the  contrary,  they  have  often  been  so 
intimately  associated  that  the  speculations  in  the  latter  have 
influenced  unduly  the  conclusions  in  the  former.  While  it  is 
clear  that  the  two  kinds  of  investigation  should.be  kept  distinct 
in  the  mind  of  the  investigator,  it  is  equally  clear  that  they  should 
be  constantly  employed  as  checks  upon  each  other.  The  object 
of  the  next  two  articles  will  be  to  trace,  in  as  few  words  as  possible, 
the  development  of  these  two  lines  of  progress  of  the  science  of 
Celestial  Mechanics  from  the  times  of  the  early  Greek  Philosophers 
to  the  time  when  Newton  applied  his  transcendent  genius  to  the 
analysis  of  the  elements  involved,  and  to  their  synthesis  into  one 
of  the  sublimest  products  of  the  human  mind. 

26.  Formal  Astronomy.  The  first  division  is  concerned  with 
phenomena  altogether  apart  from  their  causes,  and  will  be  termed 
Formal  Astronomy.  The  day,  the  month,  and  the  year  are  such 
obvious  natural  divisions  of  time  that  they  must  have  been 
noticed  by  the  most  primitive  peoples.  But  the  determination  of 
the  relations  among  these  periods  required  something  of  the  sci- 
entific spirit  necessary  for  careful  observations;  yet,  in  the  very 
dawn  of  Chaldean  and  Egyptian  history  they  appear  to  have  been 
known  with  a  considerable  degree  of  accuracy.  The  records  left 
by  these  peoples  of  their  earlier  civilizations  are  so  meager  that 
little  is  known  with  certainty  regarding  their  scientific  achieve- 
ments. The  authentic  history  of  Astronomy  actually  begins  with 
the  Greeks,  who,  deriving  their  first  knowledge  and  inspiration 
from  the  Egyptians,  pursued  the  subject  with  the  enthusiasm 
and  acuteness  which  was  characteristic  of  the  Greek  race. 

Thales  (640-546  B.C.),  of  Miletus,  went  to  Egypt  for  instruc- 
tion, and  on  his  return  founded  the  Ionian  School  of  Astronomy 
and  Philosophy.  Some  idea  of  the  advancement  made  by  the 
Egyptians  can  be  gathered  from  the  fact  that  he  taught  the 
sphericity  of  the  earth,  the  obliquity  of  the  ecliptic,  the  causes  of 
eclipses,  and,  according  to  Herodotus,  predicted  the  eclipse  of  the 
sun  of  585  B.C.  According  to  Laertius  he  was  the  first  to  deter- 
mine the  length  of  the  year.  It  is  fair  to  assume  that  he  borrowed 


26]  FROM   ANCIENT  TIMES   TO   NEWTON.  31 

much  of  his  information  from  Egypt,  though  the  basis  for  pre- 
dicting eclipses  rests  on  the  period  of  6585  days,  known  as  the 
saros,  discovered  by  the  Chaldaeans.  After  the  lapse  of  this 
period  eclipses  of  the  sun  and  moon  recur  under  almost  identical 
circumstances  except  that  they  are  displaced  about  120°  westward 
on  the  earth. 

Anaximander  (611-545  B.C.),  a  friend  and  probably  a  pupil  of 
Thales,  constructed  geographical  maps,  and  is  credited  with 
having  invented  the  gnomon. 

Pythagoras  (about  569-470  B.C.)  travelled  widely  in  Egypt  and 
Chaldea,  penetrating  Asia  even  to  the  banks  of  the  Ganges.  On 
his  return  he  went  to  Sicily  and  founded  a  School  of  Astronomy 
and  Philosophy.  He  taught  that  the  earth  both  rotates  and 
revolves,  and  that  the  comets  as  well  as  the  planets  move  in  orbits 
around  the  sun.  He  is  credited  with  being  the  first  to  maintain 
that  the  same  planet,  Venus,  is  both  evening  and  morning  star  at 
different  times. 

Meton  (about  465-385  B.C.)  brought  to  the  notice  of  the 
learned  men  of  Hellas  the  cycle  of  19  years,  nearly  equalling  235 
synodic  months,  which  has  since  been  known  as  the  Metonic  cycle. 
After  the  lapse  of  this  period  the  phases  of  the  moon  recur  on  the 
same  days  of  the  year,  and  almost  at  the  same  time  of  day.  The 
still  more  accurate  Callipic  cycle  consists  of  four  Metonic  cycles, 
less  one  day. 

Aristotle  (384-322  B.C.)  maintained  the  theory  of  the  globular 
form  of  the  earth  and  supported  it  with  many  of  the  arguments 
which  are  used  at  the  present  time. 

Aristarchus  (310-250  B.C.)  wrote  an  important  work  entitled 
Magnitudes  and  Distances.  In  it  he  calculated  from  the  time  at 
which  the  earth  is  in  quadrature  as  seen  from  the  moon  that  the 
latter  is  about  one-nineteenth  as  distant  from  the  earth  as  the  sun. 
The  time  in  question  is  determined  by  observing  when  the  moon 
is  at  the  first  quarter.  The  practical  difficulty  of  determining 
exactly  when  the  moon  has  any  particular  phase  is  the  only  thing 
that  prevents  the  method,  which  is  theoretically  sound,  from 
being  entirely  successful. 

Eratosthenes  (275-194  B.C.)  made  a  catalogue  of  475  of  the 
brightest  stars,  and  is  famous  for  having  determined  the  size  of 
the  earth  from  the  measurement  of  the  difference  in  latitude  and 
the  distance  apart  of  Syene,  in  Upper  Egypt,  and  Alexandria. 

Hipparchus  (about  190-120  B.C.),  a  native  of  Bithynia,  who 


32  HISTORICAL  SKETCH.  [26 

observed  at  Rhodes  and  possibly  at  Alexandria,  was  the  greatest 
astronomer  of  antiquity.  He  added  to  zeal  and  skill  as  an  ob- 
server the  accomplishments  of  the  mathematician.  Following 
Euclid  (about  330-275  B.C.)  at  Alexandria,  he  developed  the 
important  science  of  Spherical  Trigonometry.  He  located  places 
on  the  earth  by  their  Longitudes  and  Latitudes,  and  the  stars  by 
their  Right  Ascensions  and  Declinations.  He  was  led  by  the 
appearance  of  a  temporary  star  to  make  a  catalogue  of  1080  fixed 
stars.  He  measured  the  length  of  the  tropical  year,  the  length 
of  the  month  from  eclipses,  the  motion  of  the  moon's  nodes  and 
that  of  her  apogee;  he  was  the  author  of  the  first  solar  tables;  he 
discovered  the  precession  of  the  equinoxes,  and  made  extensive 
observations  of  the  planets.  The  works  of  Hipparchus  are  known 
only  indirectly,  his  own  writings  having  been  lost  at  the  time  of 
the  destruction  of  the  great  Alexandrian  library  by  the  Saracens 
under  Omar,  in  640  A.D. 

Ptolemy  (100-170  A.D.)  carried  forward  the  work  of  Hipparchus 
faithfully  and  left  the  Almagest  as  the  monument  of  his  labors. 
Fortunately  it  has  come  down  to  modern  times  intact  and  contains 
much  information  of  great  value.  Ptolemy's  greatest  discovery 
is  the  evection  of  the  moon's  motion,  which  he  detected  by  fol- 
lowing the  moon  during  the  whole  month,  instead  of  confining  his 
attention  to  certain  phases  as  previous  observers  had  done.  He 
discovered  refraction,  but  is  particularly  famous  for  the  system  of 
eccentrics  and  epicycles  which  he  developed  to  explain  the  apparent 
motions  of  the  planets. 

A  stationary  period  followed  Ptolemy  during  which  science  was 
not  cultivated  by  any  people  except  the  Arabs,  who  were  imitators 
and  commentators  rather  more  than  original  investigators.  In 
the  Ninth  Century  the  greatest  Arabian  astronomer,  Albategnius 
(850-929),  flourished,  and  a  more  accurate  measurement  of  the 
arc  of  a  meridian  than  had  before  that  time  been  executed  was 
carried  out  by  him  in  the  plain  of  Singiar,  in  Mesopotamia.  In 
the  Tenth  Century  Al-Sufi  made  a  catalogue  of  stars  based  on  his 
own  observations.  Another  catalogue  was  made  by  the  direction 
of  Ulugh  Beigh  (1393-1449),  at  Samarkand,  in  1433.  At  this 
time  Arabian  astronomy  practically  ceased  to  exist. 

Astronomy  began  to  revive  in  Europe  toward  the  end  of  the 
Fifteenth  Century  in  the  labors  of  Peurbach  (1423-1461), 
Waltherus  (1430-1504),  and  Regiomontanus  (1436-1476).  It 
was  given  a  great  impetus  by  the  celebrated  German  astronomer 


27]  FROM  ANCIENT  TIMES  TO  NEWTON.  33 

Copernicus  (1473-1543),  and  has  been  pursued  with  increasing 
zeal  to  the  present  time.  Copernicus  published  his  masterpiece, 
De  Revolutionibus  Orbium  Coelestium,  in  1543,  in  which  he  gave  to 
the  world  the  heliocentric  theory  of  the  solar  system.  The 
system  of  Copernicus  was  rejected  by  Tycho  Brahe  (1546-1601), 
who  advanced  a  theory  of  his  own,  because  he  could  not  observe 
any  parallax  in  the  fixed  stars.  Tycho  was  of  Norwegian  birth, 
but  did  much  of  his  astronomical  work  in  Denmark  under  the 
patronage  of  King  Frederick.  After  the  death  of  Frederick  he 
moved  to  Prague  where  he-  was  supported  the  remainder  of  his 
life  by  a  liberal  pension  from  Rudolph  II.  He  was  an  indefatigable 
and  most  painstaking  observer,  and  made  very  important  contri- 
butions to  Astronomy.  In  his  later  years  Tycho  Brahe  had 
Kepler  (1571-1630)  for  his  disciple  and  assistant,  and  it  was  by 
discussing  the  observations  of  Tycho  Brahe  that  Kepler  was  en- 
abled, in  less  than  twenty  years  after  the  death  of  his  preceptor, 
to  announce  the  three  laws  of  -planetary  motion.  It  was  from 
these  laws  that  Newton  (1642-1727)  derived  the  law  of  gravitation. 
Galileo  (1564-1642),  an  Italian  astronomer,  a  contemporary  of 
Kepler,  and  a  man  of  greater  genius  and  greater  fame,  first  applied 
the  telescope  to  celestial  objects.  He  discovered  four  satellites 
revolving  around  Jupiter,  the  rings  of  Saturn,  and  spots  on  the 
sun.  He,  like  Kepler,  was  an  ardent  supporter  of  the  heliocentric 
theory. 

27.  Dynamical  Astronomy.  By  Dynamical  Astronomy  will  be 
meant  the  connecting  of  mechanical  and  physical  causes  with 
observed  phenomena.  Formal  Astronomy  is  so  ancient  that  it  is 
not  possible  to  go  back  to  its  origin;  Dynamical  Astronomy,  on 
the  other  hand,  did  not  begin  until  after  the  time  of  Aristotle,  and 
then  real  advances  were  made  at  only  very  rare  intervals. 

Archimedes  (287-212  B.C.),  of  Syracuse,  is  the  author  of  the 
first  sound  ideas  regarding  mechanical  laws.  He  stated  correctly 
the  principles  of  the  lever  and  the  meaning  of  the  center  of  gravity 
of  a  body.  The  form  and  generality  of  his  treatment  were  im- 
proved by  Leonardo  da  Vinci  (1452-1519)  in  his  investigations 
of  statical  moments.  The  whole  subject  of  Statics  of  a  rigid  body 
involves  only  the  application  of  the  proper  mathematics  to  these 
principles. 

It  is  a  remarkable  fact  that  no  single  important  advance  was 
made  in  the  discovery  of  mechanical  laws  for  nearly  two  thousand 
years  after  Archimedes,  or  until  the  time  of  Stevinus  (1548-1620), 
4 


34  HISTORICAL   SKETCH   TO   NEWTON.  [27 

who  was  the  first,  in  1586,  to  investigate  the  mechanics  of  the 
inclined  plane,  and  of  Galileo  (1564-1642),  who  made  the  first 
important  advance  in  Kinetics.  Thus,  the  mechanical  principles 
involved  in  the  motions  of  bodies  were  not  discovered  until  rela-. 
tively  modern  times.  The  fundamental  error  in  the  speculations 
of  most  of  the  investigators  was  that  they  supposed  that  it  required 
a  continually  acting  force  to  keep  a  body  in  motion.  They  thought 
it  was  natural  for  a  body  to  have  a  position  rather  than  a  state  of 
motion.  This  is  the  opposite  of  the  law  of  inertia  (Newton's  first 
law).  This  law  was  discovered  by  Galileo  quite  incidentally  in 
the  study  of  the  motion  of  bodies  sliding  down  an  inclined  plane 
and  out  on  a  horizontal  surface.  Galileo  took  as  his  fundamental 
principle  that  the  change  of  velocity,  or  acceleration,  is  deter- 
mined by  the  forces  which  act  upon  the  body.  This  contains 
nearly  all  of  Newton's  first  two  laws.  Galileo  applied  his  principles 
with  complete  success  to  the  discovery  of  the  laws  of  falling  bodies, 
and  of  the  motion  of  projectiles.  The  value  of  his  discoveries  is 
such  that  he  is  justly  considered  to  be  the  founder  of  Dynamics. 
He  was  the  first  to  employ  the  pendulum  for  the  measurement  of 
time. 

Huyghens  (1629-1695),  a  Dutch  mathematician  and  scientist, 
published  his  Horologium  Osdllatorium  in  1675,  containing  the 
theory  of  the  determination  of  the  intensity  of  the  earth's  gravity 
from  pendulum  experiments,  the  theory  of  the  center  of  oscil- 
lation, the  theory  of  evolutes,  and  the  theory  of  the  cycloidal 
pendulum. 

Newton  (1642-1727)  completed  the  formulation  of  the  funda- 
mental principles  of  Mechanics,  and  applied  them  with  unparalleled 
success  in  the  solution  of  mechanical  and  astronomical  problems. 
He  employed  Geometry  with  such  skill  that  his  work  has  scarcely 
been  added  to  by  the  use  of  his  methods  to  the  present  day. 

After  Newton's  time,  mathematicians  soon  turned  to  the  more 
general  and  powerful  methods  of  analysis.  The  subject  of  An- 
alytical Mechanics  was  founded  by  Euler  (1707-1783)  in  his  work, 
Mechanica  sive  Motus  Scientia  (Petersburg,  1736) ;  it  was  improved 
by  Maclaurin  (1698-1746)  in  his  work,  A  Complete  System  of 
Fluxions  (Edinburgh,  1742),  and  was  highly  perfected  by  Lagrange 
(1736-1813)  in  his  Mecanique  Analytique  (Paris,  1788).  The 
Mecanique  Celeste  of  Laplace  (1749-1827)  put  Celestial  Mechanics 
on  a  correspondingly  high  plane. 


BIBLIOGRAPHY.  35 


BIBLIOGRAPHY. 

For  the  fundamental  principles  of  Mechanics  consult  Principien  der 
Mechanik  (a  history  and  exposition  of  the  various  systems  of  Mechanics 
from  Archimedes  to  the  present  time),  by  Dr.  E.  Diihring;  Vorreden  und 
Einleitungen  der  klassischen  Werken  der  Mechanik,  edited  by  the  Phil.  Soc.  of 
the  Univ.  of  Vienna;  Die  Principien  der  Mechanik,  by  Heinrich  Hertz,  Coll. 
Works,  vol.  in;  The  Science  of  Mechanics,  by  E.  Mach,  translated  by  T.  J. 
McCormack;  Principe  der  Mechanik,  by  Boltzmann;  Newton's  Laws  of  Motion, 
by  P.  G.  Tait;  Das  Princip  der  Erhaltung  der  Energie,  by  Planck;  Die  geschicht- 
liche  Entwickelung  des  Bewegungsbegriffes,  by  Lange. 

For  the  theory  of  Relativity  consult  Das  Relativitdtsprincip,  by  M.  Laue, 
and  The  Theory  of  Relativity,  by  R.  D.  Carmichael. 

For  velocity  and  acceleration  and  their  resolution  and  composition  consult 
the  first  parts  of  Dynamics  of  a  Particle,  by  Tait  and  Steele;  Legons  de  Cine- 
matique,  by  G.  Koenigs;  Cinematique  et  Mecanismes,  by  Poincare";  and  the 
works  on  Dynamics  (Mechanics)  by  Routh,  Love,  Budde,  and  Appell. 

For  the  history  of  Celestial  Mechanics  and  Astronomy  consult  Histoire  de 
V Astronomic  Ancienne  (old  work),  by  Delambre;  Astronomische  Beobachtungen 
der  Alien  (old  work),  by  L.  Ideler;  Recherches  sur  V Histoire  de  V Astronomic 
Ancienne,  by  Paul  Tannery;  History  of  Astronomy,  by  Grant;  Geschichte  der 
Mathematik  im  Alterthum  und  Mittelalter,  by  H.  Hankel;  History  of  the  Induc- 
tive Sciences  (2  vols.),  by  Whewell;  Geschichte  der-  Mathematischen  Wissen- 
schaften  (2  vols.),  by  H.  Siiter;  Geschichte  der  Mathematik  (3  vols.),  by  M. 
Cantor;  A  Short  History  of  Mathematics,  by  W.  W.  R.  Ball;  A  History  of 
Mathematics,  by  Florian  Cajori;  Histoire  des  Sciences  Mathematiques  et 
Physiques  (12  vols.),  by  M.  Maximilian  Marie;  Geschichte  der  Astronomic,  by 
R.  Wolf;  A  History  of  Astronomy,  by  Arthur  Berry;  Histoire  Abregee  de  V  Astron- 
omic, by  Ernest  Lebon. 


CHAPTER  II. 

RECTILINEAR   MOTION. 

28.  A  great  part  of  the  work  in  Celestial  Mechanics  consists  of 
the  solution  of  differential  equations  which,  in  most  problems,  are 
very  complicated  on  account  of  the  number  of  dependent  variables 
involved.     The  ordinary  Calculus  is  devoted,  in  a  large  part,  to 
the  treatment  of  equations  in  which  there  is  but  one  independent 
variable  and  one  dependent  variable ;  and  the  step  to  simultaneous 
equations  in  several  variables,  requiring  interpretation  in  con- 
nection  with   physical   problems   and   mechanical   principles,   is 
one  which  is  usually  made  not  without  some  difficulty.     The 
present  chapter  will  be  devoted  to  the  formulation  and  to  the 
solution  of  certain  classes  of  problems  in  which  the  mathematical 
processes  are  closely  related  to  those  which  are  expounded  in  the 
mathematical  text-books.     It  will  form  the  bridge  leading  from 
the  methods  which  are  familiar  in  works  on  Calculus  and  ele- 
mentary Differential  Equations  to  those  which  are  characteristic 
of  mechanical  and  astronomical  problems. 

The  examples  chosen  to  illustrate  the  principles  are  taken 
largely  from  astronomical  problems.  They  are  of  sufficient 
interest  to  justify  their  insertion,  even  though  they  were  not 
needed  as  a  preparation  for  the  more  complicated  discussions  which 
will  follow.  They  embrace  the  theory  of  falling  bodies,  the  velocity 
of  escape,  parabolic  motion,  and  the  meteoric  and  contraction 
theories  of  the  sun's  heat. 

THE  MOTION  OF  FALLING  PARTICLES. 

29.  The  Differential  Equation  of  Motion.     Suppose  the  mass 
of  the  particle  is  m  and  let  s  represent  the  line  in  which  it  falls. 
Take  the  origin  0  at  the  surface  of  the  earth  and  let  the  positive 
end  of  the  line  be  directed  upward.     By  the  second  law  of  motion 
the  rate  of  change  of  momentum,  or  the  product  of  the  mass  and 
the  acceleration,  is  proportional  to  the  force.     Let  k2  represent  the 
factor  of  proportionality,  the  numerical  value  of  which  will  depend 

36 


30]  CASE   OF   CONSTANT  FORCE.  37 


upon  the  units  employed.  Then,  if  /  represents  the  force,  the  dif- 
ferential equation  of  motion  is 

(1)  m  g  =  -  ty. 

This  is  also  the  differential  equation  of  motion  for  any  case  in 
which  the  resultant  of  all  the  forces  is  constantly  in  the  same 
straight  line  and  in  which  the  body  is  not  initially  projected  from 
that  line.  A  more  general  treatment  will  therefore  be  given  than 
would  be  required  if  /  were  simply  the  force  arising  from  the 
earth's  attraction  for  the  particle  m. 

The  force  /  will  depend  in  general  upon  various  things,  such  as 
the  position  of  m,  the  time  t,  and  the  velocity  v.  This  may  be 
indicated  by  writing  equation  (1)  in  the  form 

(2)  m^  =  -Vf(8,t,v), 

in  which  /  (s,  t,  v)  simply  means  that  the  force  may  depend  upon 
the  quantities  contained  in  the  parenthesis.  In  order  to  solve 
this  equation  two  integrations  must  be  performed,  and  the  char- 
acter of  the  integrals  will  depend  upon  the  manner  in  which  / 
depends  upon  s,  t,  and  v.  It  is  necessary  to  discuss  the  different 
cases  separately. 

30.  Case  of  Constant  Force.  This  simplest  case  is  nearly 
realized  when  particles  fall  through  small  distances  near  the  earth's 
surface  under  the  influence  of  gravity.  If  the  second  is  taken 
as  the  unit  of  time  and  the  foot  as  the  unit  of  length  then  k2f  =  mg, 
where  g  is  the  acceleration  of  gravity  at  the  surface  of  the  earth. 
Its  numerical  value,  which  varies  somewhat  with  the  latitude, 
is  a  little  greater  than  32.  Then  equation  (1)  becomes 


This  equation  gives  after  one  integration 

ds 

f  --**  +  *• 

where  c\  is  the  constant  of  integration.  Let  the  velocity  of  the 
particle  at  the  time  t  =  0  be  v  =  VQ.  Then  the  last  equation 
becomes  at  t  =  0 

v0  =  ci; 


38  ATTRACTIVE  FORCE  VARYING  AS  DISTANCE.  [31 

whence 

ds 

-dt  =  ~  gt  +  VQ' 

The  integral  of  this  equation  is 

2 

vQt  +  c2. 


Suppose  the  particle  is  started  at  the  distance  SQ  from  the  origin 
at  the  time  t  =  0;  then  this  equation  gives 

SQ  =  c2; 
whence 

«*2 

(4) 

When  the  initial  conditions  are  given  this  equation  determines  the 
position  of  the  particle  at  any  time;  or,  it  determines  the  time  at 
which  the  body  has  any  position  by  the  solution  of  the  quadratic 
equation  in  t. 

If  the  acceleration  were  any  other  positive  or  negative  constant 
than  —  mg,  the  equation  for  the  space  described  would  differ  from 
(4)  only  in  the  coefficient  of  P. 

It  is  also  possible  to  obtain  an  important  relation  between  the 
speed  and  the  position  of  the  particle.  Multiply  both  members 

ds  f  ds  \ 2 

of  equation  (3)  by  2  -=- .     Then,  since  the  derivative  of  (  -r  )    is 

2  rfi  ^7/2 '  ^e  m^e&ral  °f  ^ne  equation  is 


1 =  ~ 2gs  + 

It  follows  from  the  initial  conditions  that 

c3  =  vQ2  -f 
whence 

/CN  /ds\2 

(5)  (dt) 

31.  Attractive  Force  Varying  Directly  as  the  Distance.  Another 
simple  case  is  that  in  which  the  force  varies  directly  as  the  distance 
from  the  origin.  Suppose  it  is  always  attractive  toward  the 
origin.  This  has  been  found  by  experiment  to  be  very  nearly  the 
law  of  tension  of  an  elastic  string  within  certain  limits  of  stretching. 
Then  the  velocity  will  decrease  when  the  particle  is  on  the  positive 
side  of  the  origin;  therefore  for  these  positions  of  the  particle  the 


31]  ATTRACTIVE   FORCE   VARYING   AS   DISTANCE.  39 

acceleration  must  be  taken  with  the  negative  sign,  and  the  differ- 
ential equation  for  positive  values  of  s  is 

/72Q 

(6)  ^=-^ 

For  positions  of  the  particle  in  the  negative  direction  from  the 
origin  the  velocity  increases  with  the  time,  and  therefore  the 
acceleration  is  positive.  The  right  member  of  equation  (6)  must 
be  taken  with  such  a  sign  that  it  will  be  positive.  Since  s  is 
negative  in  the  region  under  consideration  the  negative  sign  must 
be  prefixed,  and  the  equation  remains  as  before.  Equation  (6)  is, 
therefore,  the  general  differential  equation  of  motion  for  a  body 
subject  to  an  attractive  force  varying  directly  as  the  distance. 

ds 
Multiply  both  members  of  equation  (6)  by  2  JT  and  integrate; 

the  result  is 


ds 

If  s  =  SQ  and   -•-  =  0,   at  the  time  t  =  0,   then  this  equation 
at 

becomes 


which  may  be  written,  as  is  customary  in  separating  the  variables, 

ds  kdt 


M  - 
The  integral  of  this  equation  is 

.  s  kt 

sin"1  —  =  ±  —=  +  02. 
so  \'m 

It  is  found  from  the  initial  conditions  that  c2  =  —  ;  whence 

.     ,  s  kt    .  TT 

sin"1  —  =  ±  — —  +  -  . 

On  taking  the  sine  of  both  members,  this  equation  becomes 

(7) 

From  this  equation  it  is  seen  that  the  motion  is  oscillatory  and 
symmetrical  with  respect  to  the  origin,  with  a  period  of    ^    m . 

rC 


40  PROBLEMS. 

For  this  reason  it  is  called  the  equation  for  harmonic  motion. 

ds 
Obviously  -r  vanishes  at  some  time  during  the  motion  for  all 

initial  conditions,  and  there  was  no  real  restriction  of  the  gener- 
ality of  the  problem  in  supposing  that  it  was  zero  at  t  =  0. 

Equation  (6)  is  in  form  the  differential  equation  for  many  physi- 
cal problems.  When  the  initial  conditions  are  assigned,  it  defines 
the  motion  of  the  simple  pendulum,  the  oscillations  of  the  tuning 
fork  and  most  musical  instruments,  the  vibrations  of  a  radiating 
body,  the  small  variations  in  the  position  of  the  earth's  axis,  etc. 
For  this  reason  the  method  of  finding  its  solution  and  the  deter- 
mination of  the  constants  of  integration  should  be  thoroughly 
mastered. 

m.     PROBLEMS. 

1.  A  particle  is  started  with  an  initial  velocity  of  20  meters  per  sec.  and 
is  subject  to  an  acceleration  of  20  meters  per  sec.  What  will  be  its  velocity 
at  the  end  of  4  sees.,  and  how  far  will  it  have  moved? 

=  100  meters  per  sec. 


r 

u 


Ans. 

240  meters. 

2.  A  particle  starting  with  an  initial  velocity  of  10  meters  per  sec.  and 
moving  with  a  constant  acceleration  describes  2050  meters  hi  5  sees.     What 
is  the  acceleration? 

Ans.    a.  —  160  meters  per  sec. 

3.  A  particle  is  moving  with  an  acceleration  of  5  meters  per  sec.     Through 
what  space  must  it  move  in  order  that  its  velocity  shall  be  increased  from 
10  meters  per  sec.  to  20  meters  per  sec.? 

Ans.    30  meters. 

4.  A  particle  starting  with  a  positive  initial  velocity  of  10  meters  per  sec. 
and  moving  under  a  positive  acceleration  of  20  meters  per  sec.  described  a 
space  of  420  meters.     What  time  was  required? 

Ans.     t  =  6  sees. 

5.  Show  that,  if  a  particle  starting  from  rest  moves  subject  to  an  attractive 
force  varying  directly  as  the  initial  distance,  the  time  of  falling  from  any 
point  to  the  origin  is  independent  of  the  distance  of  the  particle. 

6.  Suppose  a  particle  moves  subject  to  an  attractive  force  varying  directly 
as  the  distance,  and  that  the  acceleration  at  a  distance  of  1  meter  is  1  meter 
a  sec.     If  the  particle  starts  from  rest  how  long  will  it  take  it  to  fall  from  a 
distance  of  20  meters  to  10  meters? 

Ans.     1.0472  sees. 

7.  Suppose  a  particle  moves  subject  to  a  force  which  is  repulsive  from 
the  origin  and  which  varies  directly  as  the  distance;  show  that  if  v  =  0  and 
s  —  So  when  t  =  0,  then 


32]  SOLUTION   OF   LINEAR   EQUATIONS  BY   EXPONENTIALS.          41 


+  Vs2  -  so2  \         K 


k 

whence,  letting  —-=.  =  K, 
Vra 

s  =~  (eKt  +  e-**)  =  so  cosh  KL 
Observe  that  equation  (7)  may  be  written  in  the  similar  form 

s  =  ~  (e*/-LK  _|-  e-*/-i*<)  =  So  C08  KI 

8.  The  surface  gravity  of  the  sun  is  28  times  that  of  the  earth.  If  a  solar 
prominence  100,000  miles  high  was  produced  by  an  explosion,  what  must  have 
been  the  velocity  of  the  material  when  it  left  the  surface  of  the  sun? 

Ans.     184  miles  per  sec. 

32.  Solution  of  Linear  Equations  by  Exponentials.  The  differ- 
ential equation  (6)  and  the  corresponding  one  for  a  repulsive  force 
are  linear  in  s  with  constant  coefficients.  There  is  a  general 
theory  which  shows  that  all  linear  equations  having  constant 
coefficients  can  be  solved  in  terms  of  exponentials;  or,  in  certain 
special  cases,  in  terms  of  exponentials  multiplied  by  powers  of  the 
independent  variable  t.  This  method  has  not  only  the  advantage 
of  generality,  but  is  also  very  simple,  and  it  will  be  illustrated  by 
solving  (5).  Consider  the  two  forms 


Assume   s  =  e™    and   substitute   in   the   differential    equations; 
whence 

XV  +  fcV  =  0, 

XV  -  fcV  =  0. 

Since  these  equations  must  be  satisfied  by  all  values  of  t  in  order 
that  ex<  shall  be  a  solution,  it  follows  that 


(9) 

I  X2  -  /c2  =  0. 

Let  the  roots  of  the  first  equation  be  Xi  and  X2;  then  the  first 
equation  of  (8)  is  verified  by  both  of  the  particular  solutions 
eAl*  and  eXs'.  The  general  solution  is  the  sum  of  these  two  particu- 
lar solutions,  each  multiplied  by  an  arbitrary  constant.  Precisely 


42  SOLUTION   OF  LINEAR  EQUATIONS  BY  EXPONENTIALS.          [32 

similar  results  hold  for  the  second  equation  of  (8).     On  putting 
in  the  value  of  the  roots,  the  respective  general  solutions  are 


is  =  Cle^kt  +  c2e-^kt, 
\  s  =  Cl'ekt  +  c2'e-kt, 


where  Ci,  C2,  c/,  and  c2'  are  the  constants  of  integration. 


ds 


s 
Suppose  s  =  «o,  and  -^  =  so'  when  t  '.  =  0;  therefore 


I  Sn  = 


=  Ci  -\-  C2, 


The  derivatives  of  (10)  are 


On  substituting  f  =  0  and  -r-  =  SQ,  it  follows  that 


V^Hfc-CzV- lfc  =  «o', 


Therefore 


fciV-lJfc 

I  Ci'k  —  c2' 

H( 


so'     \ 


Then  the  general  solutions  become 


(ID 


Or  these  expressions  can  be  written  in  the  form 


33]         FORCE  VARYING  INVERSELY  AS  SQUARE   OF  DISTANCE.          43 

s  =  SQ  cos  kt  +  -|-  sin  kt, 

s  =  SQ  cosh  kt  +  y-  sinh  Atf  . 

This  method  of  treatment  shows  the  close  relation  between  the  two 
problems  much  more  clearly  than  the  other  methods  of  obtaining 
the  solutions. 

33.  Attractive  Force  Varying  Inversely  as  the  Square  of  the 
Distance.  For  positions  in  the  positive  direction  from  the  origin 
the  velocity  decreases  algebraically  as  the  time  increases  whether 
the  motion  is  toward  or  from  the  origin;  therefore  in  this  region 
the  acceleration  is  negative.  Similarly,  on  the  negative  side  of 

k2 
the  origin  the  acceleration  is  positive.     Since  -r  is  always  positive 

s 

the  right  member  has  different  signs  in  the  two  cases.  For 
simplicity  suppose  the  mass  of  the  attracted  particle  is  unity. 
Then  the  differential  equation  of  motion  for  all  positions  of  the 
particle  in  the  positive  direction  from  the  origin  is 


ds 
On  multiplying  both  members  of  this  equation  by  2  -r  and  inte- 

grating, it  is  found  that 


/ds\2      2k2 

(dt)  =T 


Suppose  v  =  VQ  and  s  =  s0  when  t  =  0;  then 

2k2 

C\  =  V02  --  . 
SQ 

On  substituting  this  expression  for  Ci  in  (13),  it  is  found  that 


ds            j2/<^      2       2k2 
—  =  ±  -v/ ^  v02 . 

dt  *    S  SQ 

2k2  ds 

If  vQ2  —  -  '-  <  0  there  will  be  some  finite  distance  Si  at  which  - 
s0  dt 

will  vanish;  if  the  direction  of  motion  of  the  particle  is  such  that 
it  reaches  that  point  it  will  turn  there  and  move  in  the  opposite 

2k2  ds 

direction.     If  vQ2 =  0,  37  will  vanish  at  s  =  co  •  and  if  the 

s0  dt 

particle  moves  out  from  the  origin  toward  infinity  its  distance  will 


44  FORCE  VARYING   INVERSELY  AS   SQUARE   OF   DISTANCE.         [33 

become  indefinitely  great  as  the  velocity  approaches  zero.     If 

2k2  ds 

vQ2 >  0,  37  never  vanishes,  and  if  the  particle  moves  out 

So  dt 

from  the  origin  toward  infinity  its  distance  will  become  indefi- 
nitely great  and  its  velocity  will  not  approach  zero. 

2k2  ds 

Suppose  v02 <  0  and  that  37  =  0  when  s  =  si.     Then 

SQ  dt 

equation  (13)  gives  

^  p— —  | 

Si  —  S 


The  positive  or  negative  sign  is  to  be  taken  according  as  the 
particle  is  receding  from,  or  approaching  toward,  the  origin. 
This  equation  can  be  written  in  the  form 

sds  /o" 

-kdt, 


VSiS  -  S2  \Si 

and  the  integral  is  therefore 


Since  s  =  s0  when  t  =  0,  it  follows  that 

• :    .    Si    .       .  /2«0-  *1\ 

C2  =  —  \SiSo  —  SQ  ~h  TT  sin  L I  

2  \       si       / 

whence 

4 

4 

(15)      ' 


s  kt' 

This  equation  determines  the  time  at  which  the  particle  has  any 
position  at  the  right  of  the  origin  whose  distance  from  it  is  less 
than  si.  For  values  of  s  greater  than  Si,  and  for  all  negative  values 
of  s,  the  second  term  becomes  imaginary.  That  means  that  the 
equation  does  not  hold  for  these  values  of  the  variables;  this  was 
indeed  certain  because  the  differential  equations  (13)  and  (14) 
were  valid  only  for 

0  <  s  ^  si. 

Suppose  the  particle  is  approaching  the  origin;  then  the  negative 
sign  must  be  used  in  the  right  member  of  (15).  The  time  at 
which  the  particle  was  at  rest  is  obtained  by  putting  s  =  si  in 
(15),  and  is 


35]  THE  VELOCITY   FROM   INFINITY.  45 


The  time  required  for  the  particle  to  fall  from  s0  to  the  origin 
is  obtained  by  putting  s  =  0  in  (15),  and  is 


1      s     /  -  o      1/siVF     TT 

r2=  --^-v,,.,-.^--^  [""a    * 

The  time  required  for  the  particle  to  fall  from  rest  at  s  =  si  to 
the  origin  is 


34.  The  Height  of  Projection.     When  the  constant  Ci  has  been 
determined  by  the  initial  conditions,  equation  (13)  becomes 


It  follows  from  this  equation  that  the  speed  depends  only  on  the 
distance  of  the  particle  from  the  center  of  force  and  not  on  the 
direction  of  its  motion.  The  greatest  distance  to  which  the  particle 
recedes  from  the  origin,  or  the  height  of  projection  from  the  origin, 
is  obtained  by  putting  v  =  0,  which  gives 

1  --  1  ._£°! 

Si       s0       2k2  ' 

But  if  the  height  of  projection  is  measured  from  the  point  of 
projection  s0,  as  would  be  natural  in  considering  the  projection  of 
a  body  away  from  the  surface  of  the  earth,  the  formula  becomes 


$2   =   Si    —  So  = 


2/c2  - 


35.  The    Velocity   from   Infinity.    When   the    particle    starts 
from  s0  with  zero  velocity,  equation  (13)  becomes 


If  the  particle  falls  from  an  infinite  distance,  So  is  infinite  and  the 
equation  reduces  to 


From  the  investigations  of  Art.  34  it  follows  that,  if  the  par- 
ticle is  projected  from  any  point  s  in  the  positive  direction  with 


46  THE   ESCAPE   OF  ATMOSPHERES.  [36 

the  velocity  defined  by  (18),  it  will  recede  to  infinity.  The  law 
of  attraction  in  deriving  (18)  is  Newton's  law  of  gravitation; 
therefore  this  equation  can  be  used  for  the  computation  of  the 
velocity  which  a  particle  starting  from  infinity  would  acquire  in 
falling  to  the  surfaces  of  the  various  planets,  satellites,  and  the 
sun.  Then,  if  the  particle  were  projected  from  the  surfaces  of 
the  various  members  of  the  solar  system  with  these  respective 
velocities,  it  would  recede  to  an  infinite  distance  if  it  were  not 
acted  on  by  other  forces.  But  if  its  velocity  were  only  enough 
to  take  it  away  from  a  satellite  or  a  planet,  it  would  still  be  subject 
to  the  attraction  of  the  remaining  members  of  the  solar  system^ 
chief  of  which  is  of  course  the  sun,  and  it  would  not  in  general 
recede  to  infinity  and  be  entirely  lost  to  the  system. 

36.  Application  to  the  Escape  of  Atmospheres.  The  kinetic 
theory  of  gases  is  that  the  volumes  and  pressures  of  gases  are 
maintained  by  the  mutual  impacts  of  the  individual  molecules, 
which  are,  on  the  average,  in  a  state  of  very  rapid  motion.  The 
rarity  of  the  earth's  atmosphere  and  the  fact  that  the  pressure  is 
about  fifteen  pounds  to  the  square  inch,  serve  to  give  some  idea 
of  the  high  speed  with  which  the  molecules  move  and  of  the  great 
frequency  of  their  impacts.  The  different  molecules  do  not  all 
move  with  the  same  speed  in  any  given  gas  under  fixed  conditions ; 
but  the  number  of  those  which  have  a  rate  of  motion  different  from 
the  mean  of  the  squares  becomes  less  very  rapidly  as  the  differ- 
ences increase.  Theoretically,  in  all  gases  the  range  of  the  values 
of  the  velocities  is  from  zero  to  infinity,  although  the  extreme 
cases  occur  at  infinitely  rare  intervals  compared  to  the  others. 
Under  constant  pressure  the  velocities  are  directly  proportional 
to  the  square  root  of  the  absolute  temperature,  and  inversely  pro- 
portional to  the  square  root  of  the  molecular  weight. 

Since  in  all  gases  all  velocities  exist,  some  of  the  molecules  of 
the  gaseous  envelopes  of  the  heavenly  bodies  must  be  moving 
with  velocities  greater  than  the  velocity  from  infinity,  as  defined  in 
Art.  35.  If  the  molecules  are  near  the  upper  limits  of  the  atmos- 
phere, and  start  away  from  the  body  to  which  they  belong,  they 
may  miss  collisions  with  other  molecules  and  escape  never  to 
return*.  Since  the  kinetic  theory  of  gases  is  supported  by  very 
strong  observational  evidence,  and  since  if  it  is  true  it  is  probable 
that  some  molecules  move  with  velocities  greater  than  the  velocity 

*  This  theory  is  due  to  Johnstone  Stoney,  Trans.  Royal  Dublin  Soc.,  1892. 


36J  THE   ESCAPE   OF  ATMOSPHERES.  47 

from  infinity,  it  is  probable  that  the  atmospheres  of  all  celestial 
bodies  are  being  depleted  by  this  process;  but  in  most  cases  it  is 
an  excessively  slow  one,  and  is  compensated,  to  some  extent  at 
least,  by  the  accretion  of  meteoric  matter  and  atmospheric  mole- 
cules from  other  bodies.  In  the  upper  regions  of  the  gaseous 
envelopes,  from  which  alone  the  molecules  escape,  the  temperatures 
are  low,  at  least  for  planetary  bodies  like  the  earth,  and  high 
velocities  are  of  rare  occurrence.  If  the  mean  square  velocity 
were  as  great  as,  or  exceeded,  the  velocity  from  infinity  the  deple- 
tion would  be  a  relatively  rapid  process.  In  any  case  the  elements 
and  compounds  with  low  molecular  weights  would  be  lost  most 
.rapidly,  and  thus  certain  ones  might  freely  escape  and  others  be 
largely  retained. 

The  manner  in  which  the  velocity  from  infinity  with  respect 
to  the  surface  of  an  attracting  sphere  varies  with  its  mass  and 
radius  will  now  be  investigated.  The  mass  of  a  body  is  propor- 
tional to  the  product  of  its  density  and  cube  of  its  radius.  Then 
k2,  which  is  the  attraction  at  unit  distance,  varies  directly  as  the 
mass,  and  therefore .  directly  as  the  product  of  the  density  and 
the  cube  of  the  radius.  Hence  it  follows  from  (18)  that  the 
velocity  from  infinity  at  the  surface  of  a  body  varies  as  the  product 
of  its  radius  and  the  square  root  of  its  density.  The  densities 
and  the  radii  of  the  members  of  the  solar  system  are  usually  ex- 
pressed in  terms  of  the  density  and  the  radius  of  the  earth;  hence 
the  velocity  from  infinity  can  be  easily  computed  for  each  of  them 
after  it  has  been  determined  for  the  earth. 

Let  R  represent  the  radius  of  the  earth,  and  g  the  acceleration 
of  gravitation  at  its  surface.  Then  it  follows  that 

(19)  ,-|. 

On  substituting  the  value  of  k2  determined  from  this  equation 
into  (18),  the  expression  for  the  square  of  the  velocity  becomes 


(dt)          s    • 

ds 
Let  V  =  -37  when  s  =  R\  whence 

72  =  2flffl. 
Let  a  second  be  taken  as  the  unit  of  time,  and  a  meter  as  the  unit 


48 


THE   ESCAPE   OF  ATMOSPHERES. 


[36 


of  length.  Then  R  =  6,371,000,  and  g  =  9.8094*.  On  substi- 
tuting in  the  last  equation  and  carrying  out  the  computation,  it 
is  found  that  V  =  11,180  meters,  or  about  6.95  miles,  per  sec. 
On  taking  the  values  of  the  relative  densities  and  radii  of  the 
planets  as  given  in  Chapter  XI  of  Moulton's  Introduction  to 
Astronomy,  the  following  results  are  found : 


Body 

Velocity  of  Escape 

Earth 

11,180  meters,  or      6.95  miles,  per  sec. 

Moon 

2,396       ' 

1.49      ' 

«      (i 

Sun 

618,200       ' 

384.30      ' 

U           (( 

Mercury 

3,196       ' 

1.99      ' 

U           (( 

Venus 

10,475       ' 

6.51      * 

((       11 

Mars 

5,180       ' 

3.22      ' 

l(       11 

Jupiter 

61,120       ' 

38.04      ' 

It         « 

Saturn 

37,850       ' 

23.53      ' 

1C           11 

Uranus 

23,160       ' 

14.40      ' 

11       u 

Neptune 

20,830       " 

12.95      '         "      " 

The  velocity  from  infinity  decreases  as  the  distance  from  the 
center  of  a  planet  increases,  and  the  necessary  velocity  of  pro- 
jection in  order  that  a  molecule  may  escape  decreases  corre- 
spondingly; and  the  centrifugal  acceleration  of  the  planet's  rotation 
adds  to  the  velocities  of  certain  molecules. 

The  question  arises  whether,  under  the  conditions  prevailing 
at  the  surfaces  of  the  various  bodies  enumerated,  the  average 
molecular  velocities  of  the  atmospheric  elements  do  not  equal  or 
surpass  the  corresponding  velocity  from  infinity. 

It  is  possible  to  find  experimentally  the  pressure  exerted  by  a 
gas  having  a  given  density  and  temperature  upon  a  unit  surface, 
from  which  the  mean  square  velocity  can  be  computed.  It  is 
shown  in  the  kinetic  theory  of  gases  that  the  square  root  of  the 
mean  square  velocity  of  hydrogen  molecules  at  the  temperature 
0°  Centigrade  under  atmospheric  pressure  is  about  1,700  meters  per 
second.  Under  the  same  conditions  the  velocities  of  oxygen  and 
nitrogen  molecules  are  about  one-fourth  as  great. 

The  surface  temperatures  of  the  inferior  planets  are  certainly 
much  greater  than  zero  degrees  Centigrade  in  the  parts  where 
they  receive  the  rays  of  the  sun  most  directly,  even  if  all  the  heat 
which  may  ever  have  been  received  from  their  interiors  is  neglected. 
It  seems  probable  from  the  geological  evidences  of  igneous  action 

*  Annuaire  du  Bureau  des  Long,     g  is  given  for  the  lat.  of  Paris,  48°  50'. 


37]          FORCE  PROPORTIONAL  TO  THE  VELOCITY.  49 

upon  the  earth  that  in  the  remote  past  they  were  at  a  much  higher 
temperature,  and  the  superior  planets  have  not  yet  cooled  down 
to  the  solid  state.  There  is  evidence  that  the  most  refractory 
substances  have  been  in  a  molten  state  at  some  time,  which  implies 
that  they  have  had  a  temperature  of  3000  or  4000  degrees  Centi- 
grade. Therefore  the  square  root  of  the  mean  square  velocity 
may  have  been  much  greater  than  the  approximate  mile  a  second 
for  hydrogen  given  above,  and  it  probably  continued  much  greater 
for  a  long  period  of  time.  On  comparing  these  results  with  the 
table  of  velocities  from  infinity,  it  is  seen  that  the  moon  and 
inferior  planets,  according  to  this  theory,  could  not  possibly  have 
retained  free  hydrogen  and  other  elements  of  very  small  molecular 
weights,  such  as  helium,  in  their  envelopes;  in  the  case  of  the 
moon,  Mercury,  and  Mars,  the  escape  of  heavier  molecules  as 
nitrogen  and  oxygen  must  have  been  frequent.  This  is  especially 
probable  if  the  heated  atmospheres  extended  out  to  great  dis- 
tances. The  superior  planets,  and  especially  the  sun,  could  have 
retained  all  of  the  familiar  terrestrial  elements,  and  from  this  theory 
it  should  be  expected  that  these  bodies  would  be  surrounded  with 
extensive  gaseous  envelopes. 

The  observed  facts  are  that  the  moon  has  no  appreciable 
atmosphere  whatever;  Mercury  an  extremely  rare  one,  if  any  at 
all;  Mars,  one  much  less  dense  than  that  of  the  earth;  Venus,  one 
perhaps  somewhat  more  dense  than  that  of  the  earth;  on  the 
other  hand  the  superior  planets  are  surrounded  by  extensive 
gaseous  envelopes. 

37.  The  Force  Proportional  to  the  Velocity.  When  a  particle 
moves  in  a  resisting  medium  the  forces  to  which  it  is  subject 
depend  upon  its  velocity.  Experiments  have  shown  that  in  the 
earth's  atmosphere  when  the  velocity  is  less  than  3  meters  per 
second  the  resistance  varies  nearly  as  the  first  power  of  the  velocity; 
for  velocities  from  3  to  300  meters  per  second  it  varies  nearly  as 
the  second  power  of  the  velocity;  and  for  velocities  about  400 
meters  per  second,  nearly  as  the  third  power  of  the  velocity. 

(a)  Consider  first  the  case  where  the  resistance  varies  as  the 
first  power  of  the  velocity,  and  suppose  the  motion  is  on  the 
earth's  surface  in  a  horizontal  direction  with  no  force  acting  except 
that  arising  from  the  resistance.  Then  the  differential  equation 
of  motion  is 

(20)  +*-0- 


50  FORCE   PROPORTIONAL  TO   THE  VELOCITY.  [37 

where  k2  is  a  positive  constant  which  depends  upon  the  units 
employed,  the  nature  of  the  body,  and  the  character  of  the  resisting 
medium.     Equation  (20)  is  linear  in  the  dependent  variable  s,  and 
the  general  method  of  solving  linear  equations  can  be  applied. 
Assume  the  particular  solution 

s  =  e". 
Substitute  in  (20)  and  divide  by  ext;  then 

X2  +  k2\  =  0. 
The  roots  of  this  equation  are 

Xi  =  0, 

X2  =  -  &2, 
and  the  general  solution  is 


ds 
Suppose  -j-  =  v0  and   s  =  s0  when   t  —  0.     Then  the   constants 

Ci  and  02  can  be  determined  in  terms  of  VQ  and  s0,  and  the  solution 
becomes 

'(22)  '  «  =  «.+^-^>.         . 

(&)  Consider  the  case  where  the  resistance  varies  as  the  first 
power  of  the  velocity  and  suppose  the  motion  is  in  the  vertical  line. 
Take  the  positive  end  of  the  axis  upward.  When  the  motion  is 
upward  the  velocity  is  positive  and  the  resistance  diminishes  the 
velocity.  Therefore  when  the  motion  is  upward  the  resistance 
produces  a  negative  acceleration,  and  the  differential  equation  of 
motion  is 


When  the  motion  is  downward  the  resistance  algebraically  in- 
creases the  velocity;  therefore  in  this  case  the  resistance  produces 
a  positive  acceleration.  But  since  the  velocity  is  opposite  in 
sign  in  the  two  cases,  equation  (23)  holds  whether  the  particle  is 
ascending  or  descending. 

Equation  (23)  is  linear,  but  not  homogeneous,  and  it  can  easily 
be  solved  by  the  method  known  as  the  Variation  of  Parameters. 


37]          FORCE  PROPORTIONAL  TO  THE  VELOCITY.  51 

This  method  is  so  important  in  astronomical  problems  that  it 
will  be  introduced  in  the  present  simple  connection,  though  it  is 
not  at  all  necessary  in  order  to  obtain  the  solution  of  (23).  In 
order  to  apply  the  method  consider  first  the  equation 

(<>A\  ^+fc2^_() 

d?  H    ^  dt  ~  U' 

which  is  obtained  from  (23)  simply  by  omitting  the  term  which 
does  not  involve  s.  The  general  solution  of  this  equation  is 
the  first  of  (21).  The  method  of  the  variation  of  parameters,  or 
constants,  consists  in  so  determining  Ci  and  c2  as  functions  of  t 
that  the  differential  equation  shall  be  satisfied  when  the  right 
member  is  included.  This  imposes  only  one  condition  upon  the 
two  quantities  Ci  and  c2,  and  another  can  therefore  be  added. 

If  the  coefficients  Ci  and  c2  are  regarded  as  functions  of  t,  it 
is  found  on  differentiating  the  first  of  (21)  that 


__       9 
dt  dt  dt 


As  the  supplementary  condition  on  Ci  and  c2  these  quantities  will 
be  made  subject  to  the  relation 


which  simplifies  the  expression  for  -=-  .     Then  it  is  found  that 


(26) 

and  equation  (23)  gives 

(27)  »%  =  -  ,. 

It  follows  from  this  equation  and  (25)  that 

dci       _g_  dct 

dt  "        W  dt 

(28) 


c2, 


where  c/  and  c2r  are  new  constants  of  integration.     When  these 
values  of  Ci  and  c2  are  substituted  in  (21),  it  is  found  that 


(29)  s  =  Cl'  +  . 


52  FORCE  PROPORTIONAL  TO   THE  VELOCITY.  [37 

Since  c\   is  arbitrary  it  can  of  course  be  supposed  to  include  the 
constant  p. 

The  expression  (29)  is  the  general  solution  of  (23)  because  it 
contains  two  arbitrary  constants,  c/  and  c2',  and  when  substituted 
in  (23)  satisfies  it  identically  in  t.  It  will  be  observed  that  the 
part  of  the  solution  depending  on  c\  and  c2'  has  the  same  form 
as  the  solution  of  (20).  It  is  clear  that  the  general  solution  could 
have  been  found  by  the  same  method  if  the  right  member  of  (23) 
had  been  a  known  function  of  t,  instead  of  the  constant  —  g. 

The  velocity  of  the  particle  is  found  from  (29)  to  be  given  by 
the  equation 

(30)  =  '        • 


ds 
f  Suppose  s  =  §o,  -57  =  ^o  at  t  =  0.     On  putting  these  values  in 

equations  (29)  and  (30),  it  is  found  that 

So   =   Ci'  +  Cs'+, 


whence 


,  _        v0       g 

~P""F' 

Consequently,  when,  the  constants  are  determined  by  the  initial 
conditions,  the  general  solution  (29)  becomes 


ds 
The  particle  reaches  its  highest  point  when  -^  is  zero.     Let  T 

represent  the  time  it  reaches  this  point  and  S  —  s0  the  height  of 
this  point;  then  it  is  found  from  equations  (31)  that 

kzT  1       |_  "*P(| 

+  T' 


38]       FORCE  PROPORTIONAL  TO  SQUARE  OF  VELOCITY.       53 

38.  The  Force  Proportional  to  the  Square  of  the  Velocity.    At 

the  velocity  of  a  strong  wind,  or  of  a  body  falling  any  considerable 
distance,  or  of  a  ball  thrown,  the  resistance  varies  very  nearly  as 
the  square  of  the  velocity.  An  investigation  will  now  be  made 
of  the  character  of  the  motion  of  a  particle  when  projected  upward 
against  gravity,  and  subject  to  a  resistance  from  the  atmosphere 
varying  as  the  square  of  the  velocity.  For  simplicity  in  writing, 
the  acceleration  due  to  resistance  at  unit  velocity  will  be  taken  as 
kzg.  Then  the  differential  equation  of  motion  for  a  unit  particle  is 

<»          S-  -•-*(*)'• 

This  equation  may  be  written  in  the  form 
d 


of  which  the  integral  is 

(33)  tan-i  =  -  kgt 


ds 
If  -j7  =  VQ  and  SQ  =  0  when  t  =  0,  then 

at  , 

Ci  =  tan"1  (kvo). 

On  substituting  in  (33)  and  taking  the  tangent  of  both  members, 
it  is  found  that 

,     .  ds  _  1    v0k  —  tan  (kgt) 

dt      kl  +  v0k  tan  (kgt)  ' 

This  equation  expresses  the  velocity  in  terms  of  the  time.  On 
multiplying  both  numerator  and  denominator  of  the  right  member 
of  (34)  by  cos  (kgt)j  the  numerator  becomes  the  derivative  of  the 
denominator  with  respect  to  the  time.  Then  integrating,  the 
final  solution  becomes 

(35)          s  =  T^-  log  [vok  sin  (kgt)  +  cos  (kgt)]  -f  c2. 
K  g 

It  follows  from  the  initial  conditions  that  c2  =  0.  This  equation 
expresses  the  distance  passed  over  in  terms  of  the  time. 

The  equations  can  be  so  treated  that  the  velocity  will  be  ex- 
pressed in  terms  of  the  distance.     Equation  (32)  can  be  written 


54  FORCE  PROPORTIONAL  TO   SQUARE  OF  VELOCITY.  [38 

5*_ 

of  which  the  integral  is 


From  the  initial  conditions  it  follows  that 

ci'  =  log  (1  +  &W 
Therefore 

(36) 

The  maximum  height,  which  is  reached  when  the  velocity  becomes 
zero,  is  found  from  (36)  to  be 


The  time  of  reaching  the  highest  point,  which  is  found  by  putting 

ds 

-JT  equal  to  zero  in  (34),  is  given  by 

(it 


T  =    -tan-1  (vjc). 
kg 

When  the  particle  falls  the  resistance  acts  in  the  opposite 
direction  and  the  sign  of  the  last  term  in  (32)  is  changed.  This 
may  be  accomplished  by  writing  k  V  —  1  instead  of  k,  and  if  this 
change  is  made  throughout  the  solution  the  results  will  be  valid. 
Of  course  the  results  should  be  written  in  the  exponential  form, 
instead  of  the  trigonometric  as  they  were  in  (34)  and  (35),  in  order 
to  avoid  the  appearance  of  imaginary  expressions.  If  the  initial 
velocity  is  zero,  VQ  =  Q  and  the  equations  corresponding  to  (34)  , 
(35),  and  (36)  are  repectively 


(37) 


r         ds 

1 

ekgt  _ 

g-A:0* 

dt 

(dsV 
I  (dt) 

k 

ekgt  . 

ekgt  _|_ 
f  e-kgt 

g-*»«  ' 
j 

=  AT2(1 

2 

PROBLEMS. 


55 


1.  Show  that 


IV.     PROBLEMS. 


dp  -      V? 

where  the  positive  square  root  of  s6  is  always  taken,  holds  for  the  problem  of 
Art.  33  whichever  side  of  the  origin  the  particle  may  be.  Integrate  this 
equation. 

2.  Let  s  =  s'  —  p<  in  equation  (23);  integrate  directly  and  show  that  the 
result  is  the  same  as  that  found  by  the  variation  of  parameters. 

3.  Find  equations  (37)  by  direct  integration  of  the  differential  equations. 

4.  Suppose  a  particle  starts  from  rest  and  moves  subject  to  a  repulsive 
force  varying  inversely  as  the  square  of  the  distance;  find  the  velocity  and 
time  elapsed  in  terms  of  the  space  described. 


Ans. 


So 


k\-t  =>/s2  - 


log 


5.  What  is  the  velocity  from  infinity  with  respect  to  the  sun  at  the  earth's 
distance  from  the  sun? 

Ans.     42,220  meters,  or  26.2  miles,  per  sec. 

6.  Suppose  a  particle  moves  subject  to  an  attractive  force  varying  directly 
as  the  distance,  and  to  a  resistance  which  is  proportional  to  the  speed;  solve 
the  differential  equation  by  the  general  method  for  linear  equations. 

Ans.     Let  k2  be  the  factor  of  proportionality  for  the  velocity  and  Z2  for  the 
distance.     Then  the  solutions  are 


where 


Xi  = 


-  k*  +  V/c4  -  4/2 


2 

Discuss  more  in  detail  the  form  of  the  solution  and  its  physical  meaning 
when  (a)  /c4  -  4Z2  <  0,    (6)  fc4  -  4Z2  =  0,    (c)  /c4  -  4Z2  >  0. 

7.  Suppose  that  in  addition  to  the  forces  of  problem  6  there  is  a  force  jue"'; 
derive  the  solution  by  the  method  of  the  variation  of  parameters  and  discuss 
the  motion  of  the  particle. 

8.  Develop  the  method  of  the  variation  of  parameters  for  a  linear  differ- 
ential equation  of  the  third  order. 

9.  If  fc2  =  0  equation  (23)  becomes  that  which  defines  the  motion  of  a 
freely  falling  body.     Show  that  the  limit  of  the  solution  (32)  as  /c2  approaches 
zero  is 

s  =  SQ  +  Vot  —  %gP. 


56 


PARABOLIC   MOTION. 


[39 


39.  Parabolic  Motion.  There  is  a  class  of  problems  involving 
for  their  solution  mathematical  processes  which  are  similar  to 
those  employed  thus  far  in  this  chapter,  although  the  motion  is 
not  in  a  straight  line.  On  account  of  the  similarity  in  the  analysis 
a  short  discussion  of  these  problems  will  be  inserted  here. 

Suppose  the  particle  is  subject  to  a  constant  acceleration  down- 
ward; the  problem  is  to  find  the  character  of  the  curve  described 
when  the  particle  is  projected  in  any  manner.  The  orbit  will  be 
in  a  plane  which  will  be  taken  as  the  ^-plane.  Let  the  y-axis  be 
vertical  with  the  positive  end  directed  upward.  Then  the  differ- 
ential equations  of  motion  are 


(38) 


—  =  n 
dt* 

tfy 


Since  these  equations  are  independent  of   each  other,  they  can 
be  integrated  separately,  and  give 


x  = 


»--y 


„    dx  dy 

—  0,    37  =  ^0  COS  o:,     37 
at  at 


sin  a  when  i  =  0, 


Suppose  x  =  y  =  U,   37  =  v0  cos  a,   -77  =  ^o 

where  a  is  the  angle  between  the  line  of  initial  projection  and  the 
plane  of  the  horizon,  and  ^o  is  the  speed  of  the  projection.     Then 


Fig.  6. 

the  constants  of  integration  are  found  to  be 
ai  =  #o  cos  a,        0,2  =  0, 

61  =  VQ  sin  a,         bz  =  0; 
and  therefore 


\ 


39] 


PARABOLIC   MOTION. 


57 


(39) 


X   =   VQ  COS  a   •   t, 

at2 
y  =  —  ~-  +  vQ  sin  a  •  t. 


The  equation  of  the  curve  described,  which  is  found  by  elimi- 
nating t  between  these  two  equations,  is 

(40)  y-xt***-1"^"*. 

This  is  the  equation  of  a  parabola  whose  axis  is  vertical  with  its 
vertex  upward.     It  can  be  written  in  the  form 


x  --  sin  a  cos  a       = 
Q 


y 
y 


20 


The  equation  of  a  parabola  with  its  vertex  at  the  origin  has  the 
form 

x2  =  —  2py, 

where  2p  is  the  parameter.  On  comparing  this  equation  with  the 
equation  of  the  curve  described  by  the  particle,  the  coordinates 
of  the  vertex,  or  highest  point,  are  seen  to  be 

x  =  —  sin  a.  cos  a, 
9 


The  distance  of  the  directrix  from  the  vertex  is  one-fourth  of 
the  parameter;  therefore  the  equation  of  the  directrix  is 


p 


sin2  a   ,  v<?  cos2  a 

~      ~~ 


The  square  of  the  velocity  is  found  to  be 

"=(!)'  +(*)•-»->*• 

To  find  the  place  where  the  particle  will  strike  the  horizontal 
plane  put  y  =  0  in  (40).     The  solutions  for  x  are  x  =  0  and 

2v02  .  v02  . 

x  =  --  sin  a  cos  a  =  —  sin  2a. 

g  g 

From  this  it  follows  that  the  range  is  greatest  for  a  given  initial 
velocity  if  a  =  45°.     From  (39)  the  horizontal  velocity  is  seen  to 


68  PROBLEMS. 

be  v0  cos  a;  hence  the  time  of  flight  is  — -sin  a.     Therefore,  if  the 

y 

other  initial  conditions  are  kept  fixed,  the  whole  time  of  flight 
varies  directly  as  the  sine  of  the  angle  of  elevation. 

The  angle  of  elevation  to  attain  a  given  range  is  found  by 
solving 

V  Q2 

x  =  a  =  —  sin  2a 
Q 

for  a.     If  a  >  —  there  is  no  solution.     If  a  <  -—  there  are  two 

9  9 

solutions  differing  from  the  value  for  maximum  range  (a  =  45°)  by 
equal  amounts. 

If  the  variation  in  gravity  at  different  heights  above  the  earth's 
surface,  the  curvature  of  the  earth,  and  the  resistance  of  the  air 
are  neglected,  the  investigation  above  applies  to  projectiles  near  the 
earth's  surface.  For  bodies  of  great  density  the  results  given  by 
this  theory  are  tolerably  accurate  for  short  ranges.  When  the 
acceleration  is  taken  toward  the  center  of  the  earth,  and  gravity 
is  supposed  to  vary  inversely  as  the  square  of  the  distance,  the 
path  described  by  the  particle  is  an  ellipse  with  the  center  of  the 
earth  as  one  of  the  foci.  This  will  be  proved  in  the  next  chapter. 


V.     PROBLEMS. 

1.  Prove  that,  if  the  accelerations  parallel  to  the  x  and  y-axes  are  any 
constant  quantities,  the  path  described  by  the  particle  is  a  parabola  for 
general  initial  conditions. 

2.  Find  the  direction  of  the  major  axis,  obtained  in  problem  1,  in  terms  of 
the  constant  components  of  acceleration. 

3.  Under  the  assumptions  of  Art.  39  find  the  range  on  a  line  making  an 
angle  £  with  the  z-axis. 

4.  Show  that  the  direction  of  projection  for  the  greatest  range  on  a  given 
line  passing  through  the  point  of  projection  is  in  a  line  bisecting  the  angle 
between  the  given  line  and  the  ?/-axis. 

i  5.  Show  that  the  locus  of  the  highest  points  of  the  parabolas  as  a  takes 

all  values  is  an  ellipse  whose  major  axis  is  — ,  and  minor  axis,  —-. 

6.  Prove  that  the  ^elocityi)  at  any  point  equals  that  which  the  particle 
would  have  at  the  poinfl  if  it  fell  from  the  directrix  of  the  parabola. 


40]       .  WORK  AND  ENERGY.  69 

THE  HEAT  OF  THE  SUN. 

40.  Work  and  Energy.     When  a  force  moves  a  particle  against 
any  resistance  it  is  said  to  do  work.     The  amount  of  the  work  is 
proportional  to  the  product  of  the  resistance  and  the  distance 
through  which  the  particle  is  moved.     In  the  case  of  a  free  particle 
the  resistance  comes  entirely  from  the  inertia  of  the  mass;  if  there 
is  friction  this  is  also  resistance. 

Energy  is  the  power  of  doing  work.  If  a  given  amount  of  work 
is  done  upon  a  particle  free  to  move,  the  particle  acquires  a  motion 
that  will  enable  it  to  do  exactly  the  same  amount  of  work.  The 
energy  of  motion  is  called  kinetic  energy.  If  the  particle  is  retarded 
by  friction  part  of  the  original  work  expended  will  be  used  in  over- 
coming the  friction,  and  the  particle  will  be  capable  of  doing  only 
as  much  work  as  has  been  done  in  giving  it  motion.  Until  about 
1850  it  was  generally  supposed  that  work  done  in  overcoming  fric- 
tion is  partly,  or  perhaps  entirely,  lost.  In  other  words,  it  was  be- 
lieved that  the  total  amount  of  energy  in  an  isolated  system  might 
continually  decrease.  It  was  observed,  however,  that  friction 
generates  heat,  sound,  light,  and  electricity,  depending  upon  the 
circumstances,  and  that  these  manifestations  of  energy  are  of 
the  same  nature  as  the  original,  but  in  a  different  form.  It  was 
then  proved  that  these  modified  forms  of  energy  are  in  every 
case  quantitatively  equivalent  to  the  waste  of  that  originally 
considered.  The  brilliant  experiments  of  Joule  and  others,  made 
in  the  middle  of  the  nineteenth  century,  have  established  with 
great  certainty  the  fact  that  the  total  amount  of  energy  remains 
the  same  whatever  changes  it  may  undergo.  This  principle, 
known  as  the  conservation  of  energy,  when  stated  as  holding 
throughout  the  universe,  is  one  of  the  most  far-reaching  general- 
izations that  has  been  made  in  the  natural  sciences  in  a  hundred 
years.* 

41.  Computation  of  Work.     The  amount  of  work  done  by  a 
Newtonian  force  in  moving  a  free  particle  any  distance  will  now  be 
computed.     Let  m  equal  the  mass  of  the  particle  moved,  and  k2 
a  constant  depending  upon  the  mass  of  the  attracting  body  and 
the  units  employed.     Then 


*  Herbert  Spencer  regards  the  principle  as  being  axiomatic,  and  states  his 
views  in  regard  to  it  in  his  First  Principles,  part  n.,  chap.  vi. 


60  COMPUTATION   OF  WORK.  [41 

The  right  member  is  the  force  to  which  the  particle  is  subject. 
By  Newton's  third  law  it  is  numerically  equal  to  the  reaction,  or 
the  resistance  due  to  inertia.  Then  the  work  done  in  moving 
the  particle  through  the  element  of  distance  ds  is 


d?s  ,  ,         ,TJ7 

m-jpds  =  --  —as  =  dW. 

The  work  done  in  moving  the  particle  through  the  interval  from 
so  to  Si  is  found  by  taking  the  definite  integral  of  this  expression 
between  the  limits  s0  and  si.  On  performing  the  integrations  and 
substituting  the  limits,  it  is  found  that 

1 


midsA*     m 
2\~dt)  "~2 


_ 

n 


Suppose  the  initial  velocity  is  zero;  then  the  kinetic  energy  equals 
the  work  W  done  upon  the  particle,  and 


2      dt  I  V«i 

By  hypothesis,  the  particle  has  no  kinetic  energy  on  the  start, 
and  therefore  the  power  of  doing  work  equals  the  product  of  one 
half  the  mass  and  the  square  of  the  velocity.  If  the  particle  falls 
from  infinity,  s0  is  infinite,  and  the  formula  for  the  kinetic  energy 
becomes 


,     .  m  i  dsi\2  _ 

2  \  dt  /     : 


If  the  particle  is  stopped  by  striking  a  body  when  it  reaches  the 
point  si,  its  kinetic  energy  is  changed  into  some  other  form  of 
energy  such  as  heat.  It  has  been  found  by  experiment  that  a 
body  weighing  one  kilogram  falling  425  meters*  in  the  vicinity  of 
the  earth's  surface,  under  the  influence  of  the  earth's  attraction, 
generates  enough  heat  to  raise  the  temperature  of  one  kilogram 
of  water  one  degree  Centigrade.  This  quantity  of  heat  is  called 
the  calory.  f  The  amount  of  heat  generated  is  proportional  to  the 
product  of  the  square  of  the  velocity  and  the  mass  of  the  moving 
particle.  Then,  letting  Q  represent  the  number  of  calories,  it 
follows  that 

(44)  Q  =  Cmv2. 

*  Joule  found  423.5;  Rowland  427.8.  For  results  of  experiments  and 
references  see  Preston's  Theory  of  Heat,  p.  594. 

t  One-thousandth  of  this  unit,  denned  in  using  the  gram  instead  of  the  kilo- 
gram, is  also  called  a  calory. 


42]  THE   TEMPERATURE   OF  METEORS.  61 

Let  m  be  expressed  in  kilograms  and  v  in  meters  per  second. 
In  order  to  determine  the  constant  C,  take  Q  and  m  each  equal  to 
unity;  then  the  velocity  is  that  acquired  by  the  body  falling 
through  425  meters.  It  was  shown  in  Art.  30  that,  if  the  body 
falls  from  rest, 

J    «  =  -  i<7*2, 

I    v=  -gt. 
On  eliminating  t  between  these  equations,  it  is  found  that 


In    the    units    employed    g  =  9.8094,    and    since    s  =  425    and 
v2  =  8338,  it  follows  from  (44)  that 

C- 


8338' 

Then  the  general  formula  (44)  becomes 

mv2 


(45)  Q  = 


8338 ' 


where  Q  is  expressed  in  calories  if  the  kilogram,  meter,  and  second 
are  taken  as  the  units  of  mass,  distance,  and  time. 

42.  The  Temperature  of  Meteors.  The  increase  of  temperature 
of  a  body,  when  the  proper  units  are  employed,  is  equal  to  the 
number  of  calories  of  heat  acquired  divided  by  the  product  of  the 
mass  and  the  specific  heat  of  the  substance.  Suppose  a  meteor 
whose  specific  heat  is  unity  (in  fact  it  would  probably  be  much 
less  than  unity)  should  strike  the  earth  with  any  given  velocity;  it 
is  required  to  compute  its  increase  of  temperature  if  it  took  up  all 
the  heat  generated.  The  specific  heat  has  been  taken  so  that  the 
increase  of  temperature  is  numerically  equal  to  the  number  of 
calories  generated  per  unit  mass.  Meteors  usually  strike  the  earth 
with  a  velocity  of  about  25  miles,  or  40,233  meters,  per  second. 
On  substituting  40,233  for  v  and  unity  for  m  in  (45),  it  is  found 
that  T  =  Q  =  194,134,  the  number  of  calories  generated  per  unit 
mass,  or  the  number  of  degrees  through  which  the  temperature  of 
the  meteor  would  be  raised.  As  a  matter  of  fact  a  large  part  of 
the  heat  would  be  imparted  to  the  atmosphere;  but  the  quantity 
of  heat  generated  is  so  enormous  that  it  could  not  be  expected  that 
any  but  the  largest  meteors  would  last  long  enough  to  reach  the 
earth's  surface. 

A  meteor  falling  into  the  sun  from  an  infinite  distance  would 


62  METEORIC   THEORY   OF   THE   SUN'S   HEAT.  [43 

strike  its  surface,  as  has  been  seen  in  Art.  36,  with  a  velocity  of 
about  384  miles  per  second.  The  heat  generated  would  be  there- 
fore (-V/)2,  or  236,  times  as  great  as  that  produced  in  striking  the 
earth.  Thus  it  follows  that  a  kilogram  would  generate,  in  falling 
into  the  sun,  45,815,624  calories. 

43.  The  Meteoric  Theory  of  the  Sun's  Heat.  When  it  is 
remembered  what  an  enormous  number  of  meteors  (estimated  by 
H.  A.  Newton*  as  being  as  many  as  8,000,000  daily)  strike  the 
earth,  it  is  easily  conceivable  that  enough  strike  the  sun  to  main- 
tain its  temperature.  Indeed,  this  has  been  advanced  as  a  theory 
to  account  for  the  replenishment  of  the  vast  amount  of  heat  which 
the  sun  radiates.  There  can  be  no  question  of  its  qualitative 
correctness,  and  it  only  remains  to  examine  it  quantitatively. 

Let  it  be  assumed  that  the  sun  radiates  heat  equally  in  every 
direction,  and  that  meteors  fall  upon  it  in  equal  numbers  from 
every  direction.  Under  this  assumption,  the  amount  of  heat  radi- 
ated by  any  portion  of  the  surface  will  equal  that  generated  by  the 
impact  of  meteors  upon  that  portion.  The  amount  of  heat 
received  by  the  earth  will  be  to  the  whole  amount  radiated  from 
the  sun  as  the  surface  which  the  earth  subtends  as  seen  from  the 
sun  is  to  the  surface  of  the  whole  sphere  whose  radius  is  the 
distance  from  the  earth  to  the  sun.  The  portion  of  the  sun's 
surface  which  is  within  the  cone  whose  base  is  the  earth  and  vertex 
the  center  of  the  sun,  is  to  the  whole  surface  of  the  sun  as  the 
surface  subtended  by  the  earth  is  to  the  surface  of  the  whole 
sphere  whose  radius  is  the  distance  to  the  sun.  Therefore,  the 
earth  receives  as  much  heat  as  is  radiated  by,  and  consequently 
generated  upon,  the  surface  cut  out  by  this  cone.  But  the  earth 
would  intercept  precisely  as  many  meteors  as  would  fall  upon  this 
small  area,  and  would,  therefore,  receive  heat  from  the  impact  of  a 
certain  number  of  meteors  upon  itself,  and  as  much  heat  from  the 
sun  as  would  be  generated  by  the  impact  of  an  equal  number 
upon  the  sun. 

The  heat  derived  by  the  earth  from  the  two  sources  would  be  as 
the  squares  of  the  velocities  with  which  the  meteors  strike  the 
earth  and  sun.  It  was  seen  in  Art.  42  that  this  number  is  ^i^- 
Therefore,  if  this  meteoric  hypothesis  of  the  maintenance  of  the 
sun's  heat  is  correct,  the  earth  should  receive  -^^  as  much  heat 
from  the  impact  of  meteors  as  from  the  sun.  This  is  certainly 

*  Mem.  Nat.  Acad.  of  Sd.,  vol.  i. 


44]  HELMHOLTZ'S   CONTRACTION   THEORY.  63 

millions  of  times  more  heat  than  the  earth  receives  from  meteors, 
and  consequently  the  theory  that  the  sun's  heat  is  maintained  by 
the  impact  of  meteors  is  not  tenable. 

44.  Helmholtz's  Contraction  Theory.  The  amount  of  work 
done  upon  a  particle  is  proportional  to  the  product  of  the  resistance 
overcome  by  the  distance  moved.  There  is  nothing  whatever  said 
about  how  long  the  motion  shall  take,  and  if  the  work  is  converted 
into  heat  the  total  amount  is  the  same  whether  the  particle  falls 
the  entire  distance  at  once,  or  covers  the  same  distance  by  a  suc- 
cession of  any  number  of  shorter  falls.  When  a  body  contracts 
it  is  equivalent  to  a  succession  of  very  short  movements  of  all  its 
particles  in  straight  lines  toward  the  center,  and  it  is  evident  that, 
knowing  the  law  of  density,  the  amount  of  heat  which  will  be 
generated  can  be  computed. 

In  1854  Helmholtz  applied  this  idea  to  the  computation  of  the 
heat  of  the  sun  in  an  attempt  to  explain  its  source  of  supply.  He 
made  the  supposition  that  the  sun  contracts  in  such  a  manner  that 
it  always  remains  homogeneous.  While  this  assumption  is 
certainly  incorrect,  nevertheless  the  results  obtained  are  of  great 
value  and  give  a  good  idea  of  what  doubtless  actually  takes  place 
under  contraction.  The  mathematical  part  of  the  theory  is  given 
in  the  Philosophical  Magazine  for  1856,  p.  516. 

Consider  a  homogeneous  gaseous  sphere  whose  radius  is  RQ  and 
density  a.  Let  M0  represent  its  mass.  Let  dM  represent  an 
element  of  mass  taken  anywhere  in  the  interior  or  at  the  surface 
of  the  sphere.  Let  R  be  the  distance  of  dM  from  the  center  of 
the  sphere,  and  let  M  represent  the  mass  of  the  sphere  whose  radius 
is  R.  The  element  of  mass  in  polar  coordinates  is  (Art.  21) 

(46)  dM  =  vR2  cos  (j>d(j>d8dR. 

The  element  is  subject  to  the  attraction  of  the  whole  sphere 
within  it.  As  will  be  shown  in  Chapter  IV,  the  attraction  of  the 
spherical  shell  outside  of  it  balances  in  opposite  directions  so  that 
it  need  not  be  considered  in  discussing  the  forces  acting  upon  dM. 
Every  element  in  the  infinitesimal  shell  whose  radius  is  R  is 
attracted  toward  the  center  by  a  force  equal  to  that  acting  on  dM', 
therefore  the  whole  shell  can  be  treated  at  once.  Let  dM8  repre- 
sent the  mass  of  the  elementary  shell  whose  radius  is  R.  It  is 
found  by  integrating  (46)  with  respect  to  8  and  0.  Thus 

— 

(47)  dMs  =  aR2dR     2"      f 2  cos  0d0     dO  =  ±7raR2dR. 


64  HELMHOLTZ'S   CONTRACTION   THEORY.  [44 


The  force  to  which  dM8  is  subject  is  ---  ^  —  -  .     The  element 

xc 

of  work  done  in  moving  dMs  through  the  element  of  distance  dR  is 
dWs  =  - 


The  work  done  in  moving  the  shell  from  the  distance  CR  to  R  is 
the  integral  of  this  expression  between  the  limits  CR  and  R,  or 

W   -       - 


r>2  r>  I  r^ 

K  JK  \         U 

But  M  =  ^ircrR3;  hence,  substituting  the  value  of  dMs  from  (47) 
and  representing  the  work  done  on  the  elementary  shell  by 
W8  =  dW,  it  follows  that 

dVr-J^«MP(^^l*»tt. 


=  y  TrW  ( 


The  integral  of  this  expression  from  0  to  R0  gives  the  total  amount 
of  work  done  in  the  contraction  of  the  homogeneous  sphere  from 
radius  CRo  to  RQ.  That  is, 


which  may  be  written 

r  = 

If  C  equals  infinity,  then 

(49) 

*  Q 

If  the  second  is  taken  as  the  unit  of  time,  the  kilogram  as  the 
unit  of  mass,  and  the  meter  as  the  unit  of  distance,  and  if  k2  is 
computed  from  the  value  of  g  for  the  earth,  then,  after  dividing 
W  by  —  j—  ,  the  result  will  be  numerically  equal  to  the  amount  of 
heat  in  calories  that  will  be  generated  if  the  work  is  all  trans- 
formed into  this  kind  of  energy.  The  temperature  to  which  the 
body  will  be  raised,  which  is  this  quantity  divided  by  the  product 
of  the  mass  and  the  specific  heat,  is 

T  - 


where  77  is  the  specific  heat  of  the  substance.     Or,  substituting 
(48)  in  (50),  it  is  found  that 


44]  HELMHOLTZ'S   CONTRACTION   THEORY.  65 

T_W    C-l    M0       2 
"5^"     ~~C~     ~R~Q  '8338* 

By  definition,  k2  is  the  attraction  of  the  unit  of  mass  at  unit 
distance;  therefore,  if  m  is  the  mass  of  the  earth  and  r  its  radius, 
it  follows  that 

k2m 


On  solving  for  kz  and  substituting  in  (51),  the  expression  for  T 
becomes 

T  -  3(C  ~  1}      r*     MO       2? 

5r?C       '  #o  '   m  '  8338  ' 

If  the  body  contracted  from  infinity  (C  =  »),  the  amount  of 
lieat  generated  would  be  sufficient  to  raise  its  temperature  T 
degrees  Centigrade,  where  T  is  given  by  the  equation 


5  '  77  '  R0  '  m  '  8338  ' 

Suppose  the  specific  heat  is  taken  as  unity,  which  is  that  of  water.* 
The  value  of  g  is  9.8094  and 

£  =  116,356, 

xt/o 

—  =  332,000. 

m 

On  substituting  these  numbers  in  (53)  and  reducing,  it  is  found  that 
T  =  27,268,000°  Centigrade. 

Therefore,  the  sun  contracting  from  infinity  in  such  a  way  as  to 
always  remain  homogeneous  would  generate  enough  heat  to  raise  the 
temperature  of  an  equal  mass  of  water  more  than  twenty-seven  millions 
of  degrees  Centigrade. 

If  it  is  supposed  that  the  sun  has  contracted  only  from  Neptune's 
orbit  equation  (52)  can  be  used,  which  will  give  a  value  of  T 
about  -grAnr  less.  In  any  case  it  is  not  intended  to  imply  that  it 
did  ever  contract  from  such  great  dimensions  in  the  particular 
manner  assumed;  the  results  given  are  nevertheless  significant 
and  throw  much  light  on  the  question  of  evolution  of  the  solar 
system  from  a  vastly  extended  nebula.  If  the  contraction  of  the 

*  No  other  ordinary  terrestrial  substance  has  a  specific  heat  so  great  as 
unity  except  hydrogen  gas,  whose  specific  heat  is  3.409.     But  the  lighter  gases 
of  the  solar  atmosphere  may  also  have  high  values. 
6 


66  HELMHOLTZ'S   CONTRACTION   THEORY.  [44 

sun  were  the  only  source  of  its  energy,  this  discussion  would  give  a 
rather  definite  idea  as  to  the  upper  limit  of  the  age  of  the  earth. 
But  the  limit  is  so  small  that  it  is  not  compatible  with  the  con- 
clusions reached  by  several  lines  of  reasoning  from  geological 
evidence,  and  it  is  utterly  at  variance  with  the  age  of  certain 
uranium  ores  computed  from  the  percentage  of  lead  which  they 
contain.  The  recent  discovery  of  enormous  sub-atomic  energies 
which  become  manifest  in  the  disintegration  of  radium  and  several 
other  substances  prove  the  existence  of  sources  of  energy  not 
heretofore  considered,  and  suggest  that  the  sun's  heat  may  be 
supplied  partly,  if  not  largely,  from  these  sources.  It  is  certainly 
unsafe  at  present  to  put  any  limits  on  the  age  of  the  sun. 

The  experiments  of  Abbott  have  shown  that,  under  the  assump- 
tion that  the  sun  radiates  heat  equally  in  every  direction,  the 
amount  of  heat  emitted  yearly  would  raise  the  temperature  of  a 
mass  of  water  equal  to  that  of  the  sun  1.44  degrees  Centigrade.  In 
order  to  find  how  great  a  shrinkage  in  the  present  radius  would 
be  required  to  generate  enough  heat  to  maintain  the  present  radi- 
ation 10,000  years,  substitute  14,400  for  T  in  (52)  and  solve  for  C. 
On  carrying  out  the  computation,  it  is  found  that 

C  =  1.000528. 

Therefore,  the  sun  would  generate  enough  heat  in  shrinking  about 
one  four-thousandth  of  its  present  diameter  to  maintain  its  present 
radiation  10,000  years. 

The  sun's  mean  apparent  diameter  is  1924",  so  a  contraction  of 
its  diameter  of  .000528  would  make  an  apparent  change  of  only 
l."0,  a  quantity  far  too  small  to  be  observed  on  such  an  object  by 
the  methods  now  in  use.  On  reducing  the  shrinkage  to  other 
units,  it  is  found  that  a  contraction  of  the  sun's  radius  of  36.8 
meters  annually  would  account  for  all  the  heat  that  is  being  radi- 
ated at  present. 

VI.     PROBLEMS. 

1.  According  to  the  recent  work  of  Abbott,  of  the  Smithsonian  Institution, 
a  square  meter  exposed  perpendicularly  to  the  sun's  rays  at  the  earth's  distance 
would  receive  19.5  calories  per  minute.  The  average  amount  received  per 
square  meter  on  the  earth's  surface  is  to  this  quantity  as  the  area  of  a  circle 
is  to  the  surface  of  a  sphere  of  the  same  radius,  or  1  to  4.  The  earth's  surface 
receives,  therefore,  on  the  average  5  calories  per  square  meter  per  minute. 
How  many  kilograms  of  meteoric  matter  would  have  to  strike  the  earth 
with  a  velocity  of  25  miles  (40,233  meters)  per  sec',  to  generate  ^^  this  amount 
of  heat? 

Ans.     .000,000,1115  kilograms. 


HISTORICAL   SKETCH.  67 

2.  How  many  pounds  would  have  to  fall  per  day  on  every  square  mile  on 
the  average?     Tons  on  the  whole  earth? 

.  (917  pounds. 

(90,300,000  tons. 

3.  Find  the  amount  of  work  done  in  the  contraction  of  any  fraction  of 
the  radius  of  a  sphere  when  the  law  of  density  is  <r  =  -     . 


Ans.     W  =  \Wm*R  =  kZ-  •         i  or  \  -of  the  work 

done  when  the  sphere  is  homogeneous. 

4.  Laplace  assumed  that  the  resistance  of  a  fluid  against  compression  is 
directly  proportional  to  its  density,  and  on  the  basis  of  this  assumption  he 
found  that  the  law  of  density  of  a  spherical  body  would  be 


Gsin 


(^) 


where  G  and  ^  are  constants  depending  on  the  material  of  which  the  body  is 
composed,  and  where  a  is  the  radius  of  the  sphere.  This  law  of  density  is  in 
harmony,  when  applied  to  the  earth,  with  a  number  of  phenomena,  such  as 
the  precession  of  the  equinoxes.  Find  the  amount  of  heat  generated  by 
contraction  from  infinite  dimensions  to  radius  RQ  of  a  body  having  the  Lapla- 
cian  law  of  density. 

5.  Find  how  much  the  heat  generated  by  the  contraction  of  the  earth 
from  the  density  of  meteorites,  3.5,  to  the  present  density  of  5.6  would  raise 
the  temperature  of  the  whole  earth,  assuming  that  the  specific  heat  is  0.2. 

Ans.     T  =  6520.5  degrees  Centigrade. 


HISTORICAL  SKETCH  AND   BIBLIOGRAPHY. 

The  laws  of  falling  bodies  under  constant  acceleration  were  investigated 
by  Galileo  and  Stevinus,  and  for  many  cases  of  variable  acceleration  by 
Newton.  Such  problems  are  comparatively  simple  when  treated  by  the 
analytical  processes  which  have  come  into  use  since  the  time  of  Newton. 
Parabolic  motion  was  discussed  by  Galileo  and  Newton. 

The  kinetic  theory  of  gases  seems  to  have  been  first  suggested  by  J.  Ber- 
nouilli  about  the  middle  of  the  18th  century,  but  it  was  first  developed  mathe- 
matically by  Clausius.  Maxwell,  Boltzmann,  and  0.  E.  Meyer  have  made 
important  contributions  to  the  subject,  and  more  recently  Burbury,  Jeans, 
and  Hilbert.  Some  of  the  principal  books  on  the  subject  are:  Risteen's 
Molecules  and  the  Molecular  Theory  (descriptive  work);  L.  Boltzmann's 
Gastheorie;  H.  W.  Watson's  Kinetic  Theory  of  Gases;  O.  E.  Meyer's  Die  Kine- 
tische  Theorie  der  Gase;  S.  H.  Burbury's  Kinetic  Theory  of  Gases;  J.  H.  Jean's 
Kinetic  Theory  of  Gases. 


68  HISTORICAL   SKETCH. 

The  meteoric  theory  of  the  sun's  heat  was  first  suggested  by  R.  Mayer. 
The  contraction  theory  was  first  announced  in  a  public  lecture  by  Helmholtz 
at  Konigsberg  Feb.  7,  1854,  and  was  published  later  in  Phil.  Mag.  1856. 
An  important  paper  by  J.  Homer  Lane  appeared  in  the  Am.  Jour,  of  Sri. 
July,  1870.  The  amount  of  heat  generated  depends  upon  the  law  of  density 
of  the  gaseous  sphere.  Investigations  covering  this  point  are  16  papers  by 
Ritter  in  Wiedemann's  Annalen,  vol.  v.,  1878,  to  vol.  xx.,  1883;  by  G.  W.  Hill, 
Annals  of  Math.,  vol.  iv.,  1888;  and  by  G.  H.  Darwin,  Phil.  Trans.,  1888.  The 
original  papers  must  be  read  for  an  exposition  of  the  subject  of  the  heat  of 
the  sun.  Sub-atomic  energies  are  discussed  in  E.  Rutherford's  Radioactive 
Substances  and  their  Radiations. 

For  evidences  of  the  great  age  of  the  earth  consult  Chamberlin  and  Salis- 
bury's Geology,  vol.  n.,  and  vol.  in.,  p.  413  et  seq.;  for  a  general  discussion  of 
the  age  of  the  earth  see  Arthur  Holmes'  The  Age  of  the  Earth. 


CHAPTER  III. 

CENTRAL  FORCES. 

45.  Central  Force.     This  chapter  will  be  devoted  to  the  dis- 
cussion of  the  motion  of  a  material  particle  when  subject  to  an 
attractive  or  repelling  force  whose  line  of  action  always  passes 
through  a  fixed  point.     This  fixed  point  is  called  the  center  of  force. 
It  is  not  implied  that  the  force  emanates  from  the  center  or  that 
there  is  but  one  force,  but  simply  that  the  resultant  of  all  the  forces 
acting  on  the  particle  always  passes  through  this  point.     The 
force  may  be  directed  toward  the  point  or  from  it,  or  part  of  the 
time  toward  and  part  of  the  time  from  it.     It  may  be  zero  at  any 
time,  but  if  the  particle  passes  through  a  point  where  the  force  to 
which  it  is  subject  becomes  infinite,  a  special  investigation,  which 
cannot  be  taken  up  here,  is  required  to  follow  it  farther.     Since 
attractive  forces  are  of  most  frequent  occurrence  in  astronomical 
and  physical  problems,  the  formulas  developed  will  be  for  this  case; 
a  change  of  sign  of  the  coefficient  of  intensity  of  the  force  for  unit 
distance  will  make  the  formulas  valid  for  the  case  of  repulsion. 

The  origin  of  coordinates  will  be  taken  at  the  center  of  force, 
and  the  line  from  the  origin  to  the  moving  particle  is  called  the 
radius  vector.  The  path  described  by  the  particle  is  called  the 
orbit.  The  orbits  of  this  chapter  are  plane  curves.  The  planes 
are  defined  by  the  position  of  the  center  of  force  and  the  line  of 
initial  projection.  The  xy-plsme  will  be  taken  as  the  plane  of  the 
orbit. 

46.  The  Law  of  Areas.     The  first  problem  will  be  to  derive  the 
general  properties  of  motion  which  hold  for  all  central  forces.     The 
first  of  these,  which  is  of  great  importance,  is  the  law  of  areas,  and 
constitutes  the  first  Proposition  of  Newton's  Prindpia.     It  is, 
if  a  particle  is  subject  to  a  central  force,  the  areas  which  are  swept 
over  by  the  radius  vector  are  proportional  to  the  intervals  of  time  in 
which  they  are  described.     The  following  is  Newton's  demonstration 
of  it* 

Let  0  be  the  center  of  force,  and  let  the  particle  be  projected 
from  A  in  the  direction  of  B  with  the  velocity  AB.  Then,  by  the 
first  law  of  motion,  it  would  pass  to  C'  in  the  first  two  units  of 


70  THE  LAW   OF  AREAS.  [46 

time  if  there  were  no  external  forces  acting  upon  it.  But  suppose 
that  when  it  arrives  at  B  an  instantaneous  force  acts  upon  it  in 
the  direction  of  the  origin  with  such  intensity  that  it  would  move 


Fig.  7. 

to  b  in  a  unit  of  time  if  it  had  no  previous  motion.  Then,  by  the 
second  law  of  motion,  it  will  move  along  the  diagonal  of  the 
parallelogram  BbCC'  to  C.  If  no  other  force  were  applied  it 
would  move  with  uniform  velocity  to  D'  in  the  next  unit  of  time. 
But  suppose  that  when  it  arrives  at  C  another  instantaneous  force 
acts  upon  it  in  the  direction  of  the  origin  with  such  intensity 
that  it  would  move  to  c  in  a  unit  of  time  if  it  had  no  previous 
motion.  Then,  as  before,  it  will  move  along  the  diagonal  of  the 
parallelogram  and  arrive  at  D  at  the  end  of  the  unit  of  time.  This 
process  can  be  repeated  indefinitely. 

The  following  equalities  among  the  areas  of  the  triangles  in- 
volved hold,  since  they  have  sequentially  equal  bases  and  altitudes : 

OAB  =  OBC'  =  OBC  =  OCD'  =  OCD  =  etc. 

Therefore,  it  follows  that  OAB  =  OBC  =  OCD  =  ODE,  etc. 
That  is,  the  areas  of  the  triangles  swept  over  in  the  succeeding 
units  of  time  are  equal ;  and,  therefore,  the  sums  of  the  areas  of  the 
triangles  described  in  any  intervals  of  time  are  proportional  to 
the  intervals. 

The  reasoning  is  true  without  any  changes  however  small  the 
intervals  of  time  may  be.  Let  the  path  be  considered  for  some 
fixed  finite  period  of  time.  Let  the  intervals  into  which  it  is  divided 
be  taken  shorter  and  shorter;  the  impulses  will  become  closer  and 
closer  together.  Suppose  the  ratio  of  the  magnitudes  of  the  impulses 
to  the  values  of  the  intervals  between  them  remains  finite;  then  the 
broken  line  will  become  more  and  more  nearly  a  smooth  curve. 
Suppose  the  intervals  of  time  approach  zero  as  a  limit;  the  suc- 
cession of  impulses  will  approach  a  continuous  force  as  a  limit,  and 


47]  ANALYTICAL   DEMONSTRATION    OF   LAW   OF  AREAS.  71 

the  broken  line  will  approach  a  smooth  curve  as  a  limit.  The  areas 
swept  over  by  the  radius  vector  in  any  finite  intervals  of  time  are 
proportional  to  these  intervals  during  the  whole  limiting  process. 
Therefore,  the  proportionality  of  areas  holds  at  the  limit,  and  the 
theorem  is  true  for  a  continuous  central  force. 

It  will  be  observed  that  it  is  not  necessary  that  the  central  force 
shall  vary  continuously.  It  may  be  attractive  and  instantaneously 
change  to  repulsion,  or  become  zero,  and  the  law  will  still  hold; 
but  it  is  necessary  to  exclude  the  case  where  it  becomes  infinite 
unless  a  special  investigation  is  made. 

The  linear  velocity  varies  inversely  as  the  perpendicular  from 
the  origin  upon  the  tangent  to  the  curve  at  the  point  of  the  moving 
particle;  for,  the  area  described  in  a  unit  of  time  is  equal  to  the 
product  of  the  velocity  and  the  perpendicular  upon  the  tangent. 
Since  the  area  described  in  a  unit  of  time  is  always  the  same,  it 
follows  that  the  linear  velocity  of  the  particle  varies  inversely  as 
the  perpendicular  from  the  origin  to  the  tangent  of  its  orbit. 

47.  Analytical  Demonstration  of  the  Law  of  Areas.  Although 
the  language  of  Geometry  was  employed  in  the  demonstration 
of  Art.  46,  yet  the  essential  elements  of  the  methods  of  the 
Differential  and  Integral  Calculus  were  used.  Thus,  in  passing 
to  the  limit,  the  process  was  essentially  that  of  expressing  the 
problem  in  differential  equations;  and,  in  insisting  upon  com- 
paring intervals  of  finite  size  when  the  units  of  measurement  were 
indefinitely  decreased,  the  process  of  integration  was  really  em- 
ployed. It  will  be  found  that  in  the  treatment  of  all  problems 
involving  variable  forces  and  motions  the  methods  are  in  essence 
those  of  the  Calculus,  even  though  the  demonstrations  be  couched 
in  geometrical  language.  It  is  perhaps  easier  to  follow  the  reason- 
ing in  geometrical  form  when  one  encounters  it  for  the  first  time; 
but  the  processes  are  all  special  and  involve  fundamental  difficulties 
which  are  often  troublesome.  On  the  other  hand,  the  develop- 
ment of  the  Calculus  is  of  the  precise  form  to  adapt  it  to  the 
treatment  of  these  problems,  and  after  its  principles  have  been 
once  mastered,  the  application  of  it  is  characterized  by  comparative 
simplicity  and  great  generality.  A  few  problems  will  be  treated 
by  both  methods  to  show  their  essential  sameness,  and  to  illustrate 
the  advantages  of  analysis. 

Let  /  represent  the  acceleration  to  which  the  particle  is  subject. 
By  hypothesis,  the  line  of  force  always  passes  through  a  fixed 
point,  which  will  be  taken  as  the  origin  of  coordinates. 


72 


ANALYTICAL   DEMONSTRATION   OF   LAW   OF  AREAS. 


[47 


Let  0  be  the  center  of  force,  and  P  any  position  of  the  particle 
whose  rectangular  coordinates  are  x  and  y,  and  whose  polar 
coordinates  are  r  and  6.  Then  the  components  of  acceleration 


Fig.  8. 

along  the  x  and  i/-axes  are  respectively  =F  /  cos  8  and  =F  /  sin  8} 
and  the  differential  equations  of  motion  are 

d2x  ,  ,x 

—  =  ^  /  cos  8  =  **  f- 

(1) 


The  negative  sign  must  be  taken  before  the  right  members  of  these 
equations  if  the  force  is  attractive,  and  the  positive  if  it  is  repulsive. 
Multiply  the  first  equation  of  (1)  by  —  y  and  the  second  one 
by  +  x  and  add.     The  result  is 

d?y         d?x      A 


On  integrating  this  expression  by  parts,  it  is  found  that 
(2)  x^-y—=h, 

where  h  is  the  constant  of  integration. 

The  integrals  of  differential  equations  generally  lead  to  im- 
portant theorems  even  though  the  whole  problem  has  not  been 
solved,  and  in  what  follows  they  will  be  discussed  as  they  are 
obtained. 

On  referring  to  Art.  16,  it  is  seen  that  (2)  may  be  written 

dy  _     dx  _    s  d8  _  ^  dA  _  , 
Xdt~ydt~rdt~      ~dt  ~      ' 

where  A  is  the  area  swept  over  by  the  radius  vector.     The  integral 
of  this  equation  is 

A  =  M  +  c. 


49]  THE   LAWS   OF  ANGULAR  AND   LINEAR  VELOCITY.  73 

which  shows  that  the  area  is  directly  proportional  to  the  time.     » 
This  is  the  theorem  which  was  to  be  proved. 

48.  Converse  of  the  Theorem  of  Areas.     By  hypothesis 

A  =  Cit  +  c2. 

On  taking  the  derivative  with  respect  to  t,  it  is  found  that 

dA 


This  equation  becomes  in  polar  coordinates 

m  -*-».  I 

and  in  rectangular  coordinates 

dy         dx      0 

xctt~ydl  =  2ci- 

The  derivative  of  this  expression  with  respect  to  t  is 

x*y      &*  -  0. 
xw~y~d¥  '  u' 

or 

^.^_a..|/ 
dP  'd?  ~       'y' 

That  is,  the  components  of  acceleration  are  proportional  to  the 
coordinates;  therefore,  if  the  law  of  areas  is  true  with  respect  to  a 
point,  the  resultant  of  the  accelerations  passes  through  that  point. 

Or.  since  r2  -7-  =  2ci,  it  follows  that  -n  (  r2  -=-  J  =  0.     Hence,  by 

(19),  Art.  14,  the  acceleration  perpendicular  to  the  radius  vector  is 
zero;  that  is,  the  acceleration  is  in  the  line  passing  through  the 
origin. 

49.  The   Laws   of  Angular  and  Linear  Velocity.    From  the 
expression  for  the  law  of  areas  in  polar  coordinates,  it  follows  that 

m  ^-£- 

dt  ~  r*' 

therefore,  the  angular  velocity  is  inversely  proportional  to  the  square 
of  the  radius  vector. 
The  linear  velocity  is 


74  SIMULTANEOUS   DIFFERENTIAL  EQUATIONS.  [50 

ds  _  ds  dd  _  ds  h 

dt==dedi==de72' 

Let  p  represent  the  perpendicular  from  the  origin  upon  the  tangent  ; 
then  it  is  known  from  Differential  Calculus  that 

ds  =  r* 

^sLa^-j^^ 
Hence  the  expression  for  the  linear  velocity  becomes 

(4)  ^  _  h  . 

A     p9 

therefore,  the  linear  velocity  is  inversely  proportional  to  the  per- 
pendicular from  the  origin  upon  the  tangent. 

SIMULTANEOUS  DIFFERENTIAL  EQUATIONS. 

50.  The  Order  of  a  System  of  Simultaneous  Differential 
Equations.  One  integral,  equation  (2),  of  the  differential  equations 
(1)  has  been  found  which  the  motion  of  the  particle  must  fulfill. 
The  question  is  how  many  more  integrals  must  be  found  in  order 
to  have  the  complete  solution  of  the  problem. 

The  number  of  integrals  which  must  be  found  to  completely 
solve  a  system  of  differential  equations  is  called  the  order  of  the 
system.  Thus,  the  equation 


is  of  the  nth  order,  because  it  must  be  integrated  n  times  to  be 
reduced  to  an  integral  form.     Similarly,  the  general  equation 

(6)  /.g  +  /_1*^+...+/1*?  +  /..o, 

where  /„, ,  /o  are  functions  of  x  and  t,  must  be  integrated 

n  times  in  order  to  express  x  as  a  function  of  t,  and  is  of  the  nth 
order. 

An  equation  of  the  nth  order  can  be  reduced  to  an  equivalent  system 
of  n  simultaneous  equations  each  of  the  first  order.  Thus,  to  reduce 
(6)  to  a  simultaneous  system,  let 

_  dx  _  dxi  _  dXn-z 

Xl~~dt'        Xz~~dt'  "'  ~dT' 

whence 


50] 


SIMULTANEOUS  DIFFERENTIAL  EQUATIONS. 


75 


(7) 


dx 
dt 


~dt 


dt  fn  fn  fn' 

Therefore,  these  n  simultaneous  equations,  each  of  the  first  order, 
constitute  a  system  of  the  nth  order.  An  equation,  or  a  system 
of  equations,  reduced  to  the  form  (7)  is  said  to  be  reduced  to  the 
normal  form,  and  the  system  is  called  a  normal  system. 

Two  simultaneous  equations  of  order  m  and  n  can  be  reduced 
to  a  normal  system  of  order  m  +  n.     Consider  the  equations 


(8) 


fm  dtm  + 


dt' 


^  +  /0  =  0, 
dy  , 


0, 


where  the  fi  and  the  <£t  are  functions  of  x,  y,  and  t.  By  a  sub- 
stitution similar  to  that  used  in  reducing  (6),  it  follows  that  they 
are  equivalent  to  the  normal  system 

dx 


(9) 


i-i 


dt 


xm-i  — 


dy 


which  is  of  the  order  m  +  n.  Evidently  a  similar  reduction  is 
possible  when  each  equation  contains  derivatives  with  respect  to 
both  of  the  variables,  either  separately  or  as  products. 

Conversely,  a  normal  system  of  order  n  can  in  general  be  trans- 
formed into  a  single  equation  of  order  n  with  one  dependent  variable. 
To  fix  the  ideas,  consider  the  system  of  the  second  order 


76  SIMULTANEOUS   DIFFERENTIAL   EQUATIONS.  [50 

^  _   f  (r     ,,     A 

jt  ~  j  \x)  y>  i)i 


(10) 


=  0(x,  y,  t). 


In  addition  to  these  two  equations  form  the  derivative  of  one  of 
them,  for  example  the  first,  with  respect  to  t.     The  result  is 

d?x  =  df_dx      dj_dy      df 
dt2       dx  dt  ^  dydt^  df 

dii 
If  y  and  -j-  are  eliminated  between  (10)  and  (11)  the  result  will 

be  an  equation  of  the  form 

dx 


where  F  is  a  function  of  both  x  and  -rr .     Of  course,  /  and  0  of 

CLL 

equations  (10)  may  have  such  properties  that  the  elimination  of  y 
and  -JJ-  is  very  difficult. 

If  the  normal  system  were  of  the  third  order  in  the  dependent 
variables  x,  y,  and  z,  the  first  and  second  derivatives  of  the  first 
equation  would  be  taken,  and  the  first  derivative  of  the  second  and 
third  equations.  These  four  new  equations  with  the  original 

f/?7       (1%       (1  77  {1% 

three  make  seven  from  which  y,  z,  ~ ,  -j- ,  -~ ,  and  -^  can  in 

dt     dt     dt  dt 

general  be  eliminated,  giving  an  equation  of  the  third  order  in  x 

alone.     This  process  can  be  extended  to  a  system  of  any  order. 

The  differential  equations  (1)  can  be  reduced  by  the  substitution 

,      dx      .      dy  , 
x   =  -77 ,  y   =  -77  to  the  normal  system 

dx  _     ,         dxf_  .x 

dt~X'        ~dT=    =p;r> 


dt~  dt~        Jr> 

which  is  of  the  fourth  order.  Therefore  four  integrals  must  be 
found  in  order  to  have  the  complete  solution  of  the  problem. 
The  components  of  velocity,  x'  and  y',  play  roles  similar  to  the 
coordinates  in  (12),  and,  for  brevity,  they  will  be  spoken  of  fre- 
quently in  the  future  as  being  coordinates. 


51]          REDUCTION   OF   ORDER    OF    DIFFERENTIAL   EQUATIONS.  77 


51.  Reduction  of  Order.  When  an  integral  of  a  system  of 
differential  equations  has  been  found  two  methods  can  be  followed 
in  completing  the  solution.  The  remaining  integrals  can  be  found 
from  the  original  differential  equations  as  though  none  was  already 
known;  or,  by  means  of  the  known  integral,  the  order  of  the  system 
of  differential  equations  can  be  reduced  by  one.  That  the  order 
of  the  system  can  be  reduced  by  means  of  the  known  integrals 
will  be  shown  in  the  general  case.  Consider  the  system  of  differ- 
ential equations 

d-r. 

\(Xi,     •••,    Xn,    t), 


dt 

dXji 

dt 


I    Xn,    t), 


(13) 


Suppose  an  integral 

F(XI,  xz,   -',  %n,  t)  =  constant  =c, 

has  been 'found.     Suppose  this  equation  is  solved  for  xn  in  terms 
of  xi,  •  •  • ,  xn-i,  c,  and  t.     The  result  may  be  written 

Xn   =   t(Xi,    Xz,     ••',    Xn-l,    C,    t) . 

Substitute  this  expression  for  xn  in  the  first  n  -  1  equations  of  (13) ; 
they  become 


(14) 


This  is  a  simultaneous  system  of  order  n  —  1,  and  is  independent 
of  the  variable  xn. 

It  is  apparent  from  these  theorems  and  remarks  that  the  order 
of  a  simultaneous  system  of  differential  equations  is  equal  to  the 
sum  of  the  orders  of  the  individual  equations;  that  the  equations 
can  be  written  in  several  ways,  e.  g.,  as  one  equation  of  the  nth 
order,  or  n  equations  of  the  first  order;  and  that  the  integrals  may 
all  be  derived  from  the  original  system,  or  that  the  order  may  be 


t  -**... 

•,    Xn-l,    C,    t), 
',    Xn-i,    C,    0, 

dXn-1 

dt         <P"~1^1 

78  THE  VIS   VIVA   INTEGRAL.  [52 

reduced  after  each  integral  is  found.  In  mechanical  and  physical 
problems  the  intuitions  are  important  in  suggesting  methods  of 
treatment,  so  it  is  generally  advantageous  to  use  such  variables 
that  their  geometrical  and  physical  meanings  shall  be  easily 
perceived.  For  this  reason,  it  is  generally  simpler  not  to  reduce 
the  order  of  the  problem  after  each  integral  is  found. 

VII.     PROBLEMS. 

1.  Prove  the  converse  of  the  law  of  areas  by  the  geometrical  method,  and 
show  that  the  steps  agree  with  the  analysis  of  Art.  48. 

2.  Prove  the  law  of  angular  velocity  by  the  geometrical  method. 

3.  Why  cannot  equations  (1)  be  integrated  separately? 

4.  Derive  the  law  of  areas  directly  from  equation  (2)  without  passing 
to  polar  coordinates. 

5.  Show  in  detail  that  a  normal  system  of  order  four  can  be  reduced  to 
a  single  equation  of  order  four,  and  the  converse. 

6.  Reduce  the  system  of  equations  (12)  to  one  of  the  third  order  by  means 
of  the  integral  of  areas. 

52.  The  Vis  Viva  Integral.     Suppose  the  acceleration  is  toward 
the  origin;  then  the  negative  sign  must  be  taken  before  the  right 

members  of  equations  (1).     Multiply  the  first  of  (1)  by  2  -5-  , 

//7/ 

the  second  by  2  -3-  ,  and  add.     The  result  is 

ffixdx        ffiydy_    _2f(    fa  ,      ^A 
2  d?  dt       V  dt=      ~  r  (Xdt  ^~ydt  )' 

It  follows  from  r2  =  x2  +  y2  that 

dx         dy  _     dr 
Xdt~^ydt  =    r~dt' 
therefore 

9d^xdx         d^ydy=  _    f  dr 
dP  dt  1      dt2  dt  ~        J  dt' 

Suppose  /  depends  upon  r  alone,  as  it  does  in  most  astronomical 
and  physical  problems.  Then/  =  </>(r),  and 

d2x  dx         d2y  dy  .  dr 


The  integral  of  this  equation  is 


53]  FORCE   VARYING   DIRECTLY  AS   THE   DISTANCE.  79 

When  the  form  of  the  function  <f>(r)  is  given  the  integral  on  the 
right  can  be  found.     Suppose  the  integral  is  $(r)  ;  then 

(16)  vz  =  -  2$(r)  +  c. 

If  $(r)  is  a  single-valued  function  of  r,  as  it  is  in  physical  prob- 
lems, it  follows  from  (16)  that,  if  the  central  force  is  a  function  of 
the  distance  alone,  the  speed  is  the  same  at  all  points  equally 
distant  from  the  origin.  Its  magnitude  at  any  point  depends  upon 
the  initial  distance  and  speed,  and  not  upon  the  path  described. 
Since  the  force  of  gravitation  varies  inversely  as  the  square  of  the 
distance  between  the  attracting  bodies,  it  follows  that  a  body,  for 
exaniple  a  comet,  has  the  same  speed  at  a  given  distance  from  the 
sun  whether  it  is  approaching  or  receding. 

EXAMPLES  WHERE  /  is  A  FUNCTION  OF  THE  COORDINATES  ALONE. 
53.  Force  Varying  Directly  as  the  Distance.  In  order  to  find 
integrals  of  equations  (1)  other  than  that  of  areas,  the  value  of 
f  in  terms  of  the  coordinates  must  be  known.  In  the  case  in  which 
the  intensity  of  the  force  varies  directly  as  the  distance  the  inte- 
gration becomes  particularly  simple.  Let  k2  represent  the  acceler- 
ation at  unit  distance.  Then  /  =  kzr,  and,  in  case  the  force  is 
attractive,  equations  (1)  become 


dt2 


.dt*  - 

An  important  property  of  these  equations  is  that  each  is  inde- 
pendent of  the  other,  as  the  first  one  contains  the  dependent 
variable  x  alone  and  the  second  one  y  alone.  It  is  observed,  more- 
over, that  they  are  linear  and  the  solution  can  be  found  by  the 

method  given  in  Art.  32.     If  x  =  XQ,  -rr  =  XQ  ,  y=  yQ,   IT  =  yd  at 

t  =  0,  then  the  solutions  expressed  in  the  trigonometrical  form  are 

/ 
x  =  -f-  XQ  cos  kt    +  ~Y~  sin  kt, 


(18) 


-57  =  —  kxQ  sin  kt  +  XQ  cos  kt, 

f 

y  =  +  2/o  cos  kt    +  ^r-  sin  kt, 
-jr  =  —  kyQ  sin  kt  -f-  yQ'  cos  Atf. 


80  DIFFERENTIAL   EQUATION    OF   THE   ORBIT.  [54 

The  equation  of  the  orbit  is  obtained  by  eliminating  t  between  the 
first  and  third  equations  of  (18).  On  multiplying  by  the  appro- 
priate factors  and  adding,  it  is  found  that 

(zo2/o'  —  yoXo)  sin  kt  =  k(x0y  —  yQx), 


\  (xQyof  -  yox0r)  cos  kt  =  yQ'x  -  x0'y. 


The  result  of  squaring  and  adding  these  equations  is 

W  +  2/o'  V  +  (kV  +  Zo'V  -  2(&2zo2/o  +  x*'yQ')xy 


=    x0yo   —  2/o  XQ. 

This  is  the  equation  of  an  ellipse  with  the  origin  at  the  center 
unless  xQy0f  —  2/o#o'  =  0,  when  the  orbit  degenerates  to  two  straight 
lines  which  must  be  coincident;  for,  then 

#o        2/o 

—  -.  =  —.=  constant  =  c; 
XQ       2/0 
from  which 

XQ  =  cxo,        2/0  =  cyQ'. 

In  this  case  equation  (20)  becomes 

(21)  (k*<*  +  I)(y0'x  -  xo'y?  =  Q, 

and  the  motion  is  rectilinear  and  oscillatory.  In  every  case  both 
the  coordinates  and  the  components  of  velocity  are  periodic  with 

the  period  ~r  ,  whatever  the  initial  conditions  may  be. 

K 

54.  Differential  Equation  of  the  Orbit.  The  curve  described 
by  the  moving  particle,  independently  of  the  manner  in  which  it 
may  move  along  this  curve,  is  of  much  interest.  A  general  method 
of  finding  the  orbit  is  to  integrate  the  differential  equations  and  then 
to  eliminate  the  time.  This  is  often  a  complicated  process,  and  the 
question  arises  whether  the  time  cannot  be  eliminated  before  the 
integration  is  performed,  so  that  the  integration  will  lead  directly 
to  the  orbit.  It  will  be  shown  that  this  is  the  case  when  the  force 
does  not  depend  upon  the  time. 

The  differential  equations  of  motion  are  [Art.  47] 


(22) 


d?x  _       fx 
dt2  =        }  r  ' 


t2  "  r 

Since  /  does  not  involve  the  time  t  enters  only  in  the  derivatives. 


54]  DIFFERENTIAL   EQUATION    OF   THE    ORBIT.  81 

But  a  second  differential  quotient  cannot  be  separated  as  though 
it  were  an  ordinary  fraction;  therefore,  the  order  of  the  derivatives 
must  be  reduced  before  the  direct  elimination  of  t  can  be  made. 
In  order  to  do  this  most  conveniently  polar  coordinates  will  be 
employed.  Equations  (22)  become  in  these  variables 


#0 


^       r(de\- 

dt*~  r\Tt) 


__ 

dtdt 


The  integral  of  the  second  of  these  equations  is 

»dO  L 

r2-77  =  h.~   — 
di  ™>         rr 

On  eliminating  -7-  from  the  first  of  (23)  by  means  of  this  equation, 
dt 

the  result  is  found  to  be 

(24)  §  =  ?->• 

Now  let  r  =  -  ;  therefore 
u 


dr 

1  du 

I  du  d0           ,  dw 

dt 

u2  dt 
7  d  (du\ 

w2  dO  dt              dO' 

,  d2u  dO 

_       ,2  2 

~        l  dp' 

When  this  value  of  the  second  derivative  of  r  is  equated  to  the 
one  given  in  (24),  it  is  found  that 

(25)  / 

This  differential  equation  is  of  the  second  order,  but  one  integral 
has  been  used  in  determining  it;  therefore  the  problem  of  finding 
the  path  of  the  body  is  of  the  third  order.  The  complete  problem 
was  of  the  fourth  order;  the  fourth  integral  expresses  the  relation 
between  the  coordinates  and  the  time,  or  defines  the  position  of 
the  particle  in  its  orbit. 

Since  the  integral  of  (25)  expresses  u,  and  therefore  r,  in  terms 
of  6,  the  equation 

"$=>• 

when  integrated,  gives  the  relation  between  6  and  t. 

7 


82  NEWTON'S  LAW  OF  GRAVITATION.  [55 

Conversely,  equation  (25)  can  be  used  to  find  the  law  of  central 
force  which  will  cause  a  particle  to  describe  a  given  curve.  It  is 
only  necessary  to  write  the  equation  of  the  curve  in  polar  coordi- 
nates and  to  compute  the  right  member  of  (25) .  This  is  generally 
a  simpler  process  than  the  reverse  one  of  finding  the  orbit  when 
the  law  of  force  is  given. 

55.  Newton's  Law  of  Gravitation*  In  the  early  part  of  the 
seventeenth  century  Kepler  announced  three  laws  of  planetary 
motion,  which  he  had  derived  from  a  most  laborious  discussion 
of  a  long  series  of  observations  of  the  planets,  especially  of  Mars. 
They  are  the  following: 

LAW  I.  The  radius  vector  of  each  planet  with  respect  to  the  sun 
as  the  origin  sweeps  over  equal  areas  in  equal  times. 

LAW  II.  The  orbit  of  each  planet  is  an  ellipse  with  the  sun  at  one 
of  its  foci. 

LAW  III.  The  squares  of  the  periods  of  the  planets  are  to  each 
other  as  the  cubes  of  the  major  semi-axes  of  their  respective  orbits. 

It  was  on  these  laws  that  Newton  based  his  demonstration  that 
the  planets  move  subject  to  forces  directed  toward  the  sun,  and 
varying  inversely  as  the  squares  of  their  distances  from  the  sun. 
The  Newtonian  law  will  be  derived  here  by  employing  the  analyti- 
cal method  instead  of  the  geometrical  methods  of  the  Principia* 

From  the  converse  of  the  theorem  of  areas  and  Kepler's  first  law, 
it  follows  that  the  planets  move  subject  to  central  forces  directed 
toward  the  sun.  The  curves  described  are  given  by  the  second 
law,  and  equation  (25)  can,  therefore,  be  used  to  find  the  expression 
for  the  acceleration  in  terms  of  the  coordinates.  Let  a  represent 
the  major  semi-axis  of  the  ellipse,  and  e  its  eccentricity;  then  its 
equation  in  polar  coordinates  with  origin  at  a  focus  is 


1  -+-  e  cos  6 
Therefore 

tfu=         1 

w     ~T          7  /\O 


dd2      a(l  -  e2) ' 

On  substituting  this  expression  in  (25),  it  is  found  that  the  equation 
for  the  acceleration  is 

h2          I  _  *L2 
*  ~  o(l  -  e2)  '  r2  ~  r2 ' 

*  Book  i.,  Proposition  xi. 


55]  NEWTON'S  LAW  OF  GRAVITATION.  83 

Therefore,  the  acceleration  to  which  any  planet  is  subject  varies 
inversely  as  the  square  of  its  distance  from  the  sun. 

If  the  distance  r  is  eliminated  by  the  polar  equation  of  the  conic 
the  expression  for  /  has  the  form 

/  =  fci2(l  +  6  cos  0)2, 

which  depends  only  upon  the  direction  of  the  attracted  body  and 
not  upon  its  distance.  Now  for  points  on  the  ellipse  the  two 
expressions  for  /  give  the  same  value,  but  elsewhere  they  give 
different  values.  It  is  clear  that  many  other  laws  of  force,  all 
agreeing  in  giving  the  same  numerical  values  of  /  for  points  on  the 
ellipse,  can  be  obtained  by  making  other  uses  of  the  equation  of 
the  conic  to  eliminate  r.  For  example,  since  it  follows  from  the 
polar  equation  of  the  ellipse  for  points  on  its  circumference  that 
(1  +  e  cos  0)r  =  1 

a(l  -  e2) 
one  such  law  is 

+  e  cos 


~         a(\-  e2) 

and  this  value  of  /,  which  depends  both  upon  the  direction  and 
distance  of  the  attracted  body,  differs  from  both  of  the  preceding 
for  points  not  on  the  ellipse.  All  of  these  laws  are  equally  con- 
sistent with  the  motion  of  the  planet  in  question  as  expressed  by 
Kepler's  laws.  But  the  laws  of  Kepler  hold  for  each  of  the  eight 
planets  and  the  twenty-six  known  satellites  of  the  solar  system, 
besides  for  more  than  seven  hundred  small  planets  which  have  so 
far  been  discovered.  It  is  natural  to  impose  the  condition,  if  pos- 
sible, that  the  force  shall  vary  according  to  the  same  law  for  each 
body.  Since  the  eccentricities  and  longitudes  of  the  perihelia  of 
their  orbits  are  all  different,  the  law  of  force  is  the  same  for  all 
these  bodies  only  when  it  has  the  form 

W 
J  ~  rf 

Another  reason  for  adopting  this  expression  for  /  is  that  in  case  of 
all  the  others  the  attraction  would  depend  upon  the  direction  of 
the  attracted  body,  and  this  seems  improbable.  This  conclusion 
is  further  supported  by  the  fact  that  the  forces  to  which  comets 
are  subject  when  they  move  through  the  entire  system  of  planets 
vary  according  to  this  law.  And  finally,  as  will  be  shown  in  Art. 
89,  the  accelerations  to  which  the  various  planets  are  subject  vary 
from  one  to  another  according  to  this  law. 


84  EXAMPLES   OF   FINDING   THE   LAW   OF   FORCE.  [56 

From  the  consideration  of  Kepler's  laws,  the  gravity  at  the 
earth's  surface,  and  the  motion  of  the  moon  around  the  earth, 
Newton  was  led  to  the  enunciation  of  the  Law  of  Universal 
Gravitation,  which  is,  every  two  particles  of  matter  in  the  universe 
attract  each  other  with  a  force  which  acts  in  the  line  joining  them,  and 
whose  intensity  varies  as  the  product  of  their  masses  and  inversely  as 
the  squares  of  their  distance  apart. 

It  will  be  observed  that  the  law  of  gravitation  involves  con- 
siderably more  than  can  be  derived  from  Kepler's  laws  of  planetary 
motion;  and  it  was  by  a  master  stroke  of  genius  that  Newton 
grasped  it  in  its  immense  generality,  and  stated  it  so  exactly  that 
it  has  stood  without  change  for  more  than  200  years.  When 
contemplated  in  its  entirety  it  is  one  of  the  grandest  conceptions 
in  the  physical  sciences. 

56.  Examples  of  Finding  the  Law  of  Force,  (a)  If  a  particle 
describes  a  circle  passing  through  the  origin,  the  law  of  force 
(depending  on  the  distance  alone)  under  which  it  moves  is  a  very 
simple  expression.  Let  a  represent  the  radius;  then  the  polar 
equation  of  the  circle  is 

r  =  2a  cos  6,        u  —  ^—  '-  —  -  . 
2a  cos  B 

Therefore 


On  substituting  this  expression  in  (25),  it  is  found  that 

8a2h2  _  k2 
J  '-      r5      -  rs  • 

(6)  Suppose  the  particle  describes  an  ellipse  with  the  origin  at 
the  center.  The  polar  equation  of  an  ellipse  with  the  center  as 
origin  is 


_  _ 

1  -  e2  cos2  B 
From  this  it  follows  that 


bu  =  Vl  -  e2  cos2  6, 
d2u  _  e2  cos2  e  -  e2  sin2  0        e4  sin2  6  cos2  6 


U 


dB2  '        A/1  -  e2cos20  (1  ~  g2  cog2  *)f ' 

d2u      I  -  e2     1 


57]  DOUBLE   STAK   ORBITS.  85 

On  substituting  in  (25),  the  expression  for/  is  found  to  be 


THE  UNIVERSALITY  OF  NEWTON'S  LAW. 

57.  Double  Star  Orbits.  The  law  of  gravitation  is  proved 
from  Kepler's  laws  and  certain  assumptions  as  to  its  uniqueness 
to  hold  in  the  solar  system;  the  question  whether  it  is  actually 
'universal  naturally  presents  itself.  The  fixed  stars  are  so  remote 
that  it  is  not  possible  to  observe  planets  revolving  around  them, 
if  indeed  they  have  such  attendants.  The  only  observations 
thus  far  obtained  which  throw  any  light  upon  the  subject  are 
those  of  the  motions  of  the  double  stars. 

Double  star  astronomy  started  about  1780  with  the  search  for 
close  stars  by  Sir  William  Herschel  for  the  purpose  of  determining 
parallax  by  the  differential  method.  A  few  years  were  sufficient 
to  show  him,  to  his  great  surprise,  that  in  some  cases  the  two  com- 
ponents of  a  pair  were  revolving  around  each  other,  and  that, 
therefore,  they  were  physically  connected  as  well  as  being  appar- 
ently in  the  same  part  of  the  sky.  The  discovery  and  measure- 
ment of  these  systems  has  been  pursued  with  increasing  interest 
and  zeal  by  astronomers.  Burnham's  great  catalogue  of  double 
stars  contains  about  13,000  of  these  objects.  The  relative  motions 
are  so  slow  in  most  cases  that  only  a  few  have  yet  completed 
one  revolution,  or  enough  of  one  revolution  so  that  the  shapes  of 
their  orbits  are  known  with  certainty.  There  are  now  about  thirty 
pairs  whose  observed  angular  motions  have  been  sufficiently  great 
to  prove,  within  the  errors  of  the  observations,  that  they  move 
in  ellipses  with  respect  to  each  other  in  such  a  manner  that  the 
law  of  areas  is  fulfilled.  In  no  case  is  the  primary  at  the  focus, 
or  at  the  center,  of  the  relative  ellipse  described  by  the  companion, 
but  it  occupies  some  other  place  within  the  ellipse,  the  position 
varying  greatly  in  different  systems. 

From  the  observations  and  the  converse  of  the  law  of  areas  it 
follows  that  the  resultant  of  the  forces  acting  upon  one  star  of  a 
pair  is  always  directed  toward  the  other.  The  law  of  variation 
of  the  intensity  of  the  force  depends  upon  the  position  in  the 
ellipse  which  the  center  of  force  occupies.  It  must  not  be  over- 
looked at  this  point  that  the  orbits  of  the  stars  are  not  observed 
directly,  but  that  it  is  their  projections  upon  the  planes  tangent 


86  LAW   OF  FORCE  IN   BINARY  STARS.  [58 

to  the  celestial  sphere  at  their  respective  places  which  are  seen. 
The  effect  of  this  sort  of  projection  is  to  change  the  true  ellipse 
into  a  different  apparent  ellipse  whose  major  axis  has  a  different 
direction,  and  one  that  is  differently  situated  with  respect  to  the 
central  star;  indeed,  it  might  happen  that  if  one  of  the  stars  was 
really  in  the  focus  of  the  true  ellipse  described  by  the  other,  the 
projection  would  be  such  as  to  make  it  lie  on  the  minor  axis  of 
the  apparent  ellipse. 

Astronomers  have  assumed  that  the  orbits  are  plane  curves  and 
that  the  apparent  departure  of  the  central  star  from  the  focus  of 
the  ellipse  described  by  the  companion  is  due  to  projection,  and 
have  then  computed  the  angle  of  the  line  of  nodes  and  the  inclina- 
tion. No  inconsistencies  are  introduced  in  this  way,  but  the 


S  ^\  Line  of 


Nodes 


Fig.  9. 

possibility  remains  that  the  assumptions  are  not  true.  The 
question  of  what  the  law  of  force  must  be  if  it  is  not  Newton's  law 
of  gravitation  will  now  be  investigated. 

58.  Law  of  Force  in  Binary  Stars.  If  the  force  varied  directly 
as  the  distance  the  primary  star  would  be  at  the  center  of  the 
ellipse  described  by  the  secondary  (Art.  53).  No  projection  would 
change  this  relative  position,  and  since  such  a  condition  has  never 
been  observed,  it  is  inferred  that  the  force  does  not  vary  directly 
as  the  distance. 

The  condition  will  now  be  imposed  that  the  curve  shall  be  a 
conic  with  general  position  for  the  origin,  and  the  expression  for 
the  central  force  will  be  found.  The  equation  of  the  general 
conic  is 

(26)  ax2  +  2bxy  +  cy2  +  2dx  +  2fy  =  0. 


58]  LAW   OF   FORCE   IN    BINARY   STARS.  87 

On  transforming  to  polar  coordinates  and  putting  r  =  -,  this 
equation  gives 


(27)    u  =  A  sin  6  +  B  cos  0  ±  VC  sin  26  +  D  cos  26  +  H, 
where 


^  _  d2  +  ag  -  /2  -  eg          u  _  d2  +  ag  +  J2  +  eg 
' 


On  differentiating  (27)  twice,  it  is  found  that 

(28) 


d?U  •       />  D  Q 

-r^  =  —  A  sm  B  —  B  cos  B 


-C2-Z)2-(Csin20+Dcos20)2-2#(Csin20+Dcos20) 


(C  sin  20  +  D  cos  20  + 
On  substituting  (27)  and  (28)  in  (25),  it  follows  that 


r2  (C  sin  20  +  D  cos  20  +  H)*  ' 
This  becomes  as  a  consequence  of  (27) 


(30) 


/I  \3' 

( A  sin  0  —  B  cos  0  j 


There  are  also  infinitely  many  other  laws,  all  giving  the  same 
values  of  /  for  points  on  the  ellipse  in  question,  which  are  obtained 
by  multiplying  these  expressions  by  any  functions  of  u  and  0 
which  are  unity  on  the  ellipse  in  virtue  of  equation  (27). 

It  does  not  seem  reasonable  to  suppose  that  the  attraction  of 
two  stars  for  each  other  depends  upon  their  orientation  in  space. 
Equation  (29)  becomes  independent  of  0  if  C  =  D  =  0,  and  (30), 
if  A  =  B  =  0.  The  first  gives 

f  -       constant 

I    =  —2  > 

and  the  second, 

f  =  ±  constant  •  r. 

The  first  is  Newton's  law,  and  the  second  is  excluded  by  the 
fact  that  no  primary  star  has  been  found  in  the  center  of  the  orbit 
described  by  the  companion.  It  is  clear  that  0  can  be  eliminated 
from  (29)  and  (30)  by  means  of  (27)  without  imposing  the  con- 


88  GEOMETRICAL   INTERPRETATION   OF   LAW   OF   FORCE.  [59 

ditions  that  A=B  =  C  =  D  =  0.  But  Griffin  has  shown* 
that  for  all  such  laws,  except  the  Newtonian,  the  force  either 
vanishes  when  the  distance  between  the  bodi'es  vanishes,  or 
becomes  imaginary  for  certain  values  of  r.  Clearly  both  of  these 
laws  are  improbable  from  a  physical  point  of  view.  Hence  it  is 
extremely  probable  that  the  law  of  gravitation  holds  throughout 
the  stellar  systems;  and  this  conclusion  is  supported  by  the  fact 
that  the  spectroscope  shows  the  stars  are  composed  of  familiar 
terrestrial  elements. 

59.  Geometrical    Interpretation    of    the    Second    Law.     The 

expression  for  the  central  force  given  in  (30)  may  be  put  in  a  very 
simple  and  interesting  form.  Let  g3h2(H2  -  C2  -  D2)  =  N,  and 

transform A  sin  6  —  B  cos  6  into  rectangular  coordinates  and 

the  original  constants;  then  (30)  becomes 

ran  f=       ^  Nr 

(dx+fy-g)*' 

The  equation  of  the  polar  of  the  point  (xf,  y')  with  respect  to 
the  general  conic  (26)  isf 

ax,xf  +  b(xiy'  +  y&')  +  cy<y'  +  d(xi  +  x')  +  f(yi  +  y')  -  g  =  Q, 

where  x\  and  y\  are  the  running  variables.  When  (x1 ',  y'}  is  the 
origin  this  equation  becomes 

(32)  dxi  +fyi-g  =  0, 

and  has  the  same  form  as  the  denominator  of  (31).  The  values 
of  x  and  y  in  (31)  are  such  that  they  satisfy  the  equation  of  the 
conic,  while  x\  and  y\  of  (32)  satisfy  the  equation  of  the  polar  line. 
They  are,  therefore,  in  general  numerically  different  from  x  and  y. 
But  the  distance  from  any  point  (x,  y)  of  the  conic  to  the  polar 
line  with  respect  to  the  origin  is  given  by  the  equation 

=  dx  +  fy  -  g 
Vd2  +  f2 

where  x  and  y  are  the  coordinates  of  points  on  the  conic.     Let 

N'-— 0—  • 
(#  +  W 
then  (31)  becomes 

*  American  Journal  of  Mathematics,  vol.  31  (1909),  pp.  62-85. 
t  Salmon's  Conic  Sections,  Art.  89. 


PROBLEMS.  89 


Therefore,  if  a  particle  moving  subject  to  a  central  force  describes  any 
conic,  the  intensity  of  the  force  varies  directly  as  the  distance  of  the 
particle  from  the  origin,  and  inversely  as  the  cube  of  its  distance  from 
the  polar  of  the  origin  with  respect  to  the  conic. 

60.  Examples  of  Conic  Section  Motion,     (a)  When  the  orbit  is 
a  central  conic  'with  the  origin  at  the  center,  the  polar  line  recedes 

N' 
to  infinity,  and  -j  must  be  regarded  as  a  constant.     Then  the 

force  varies  directly  as  the  distance,  as  was  shown  in  Art.  56  (7>). 
(6)  When  the  origin  is  at  one  of  the  foci  of  the  conic  the  polar 

line  is  the  directrix,  and  p  =  -  ,  where  e  is  the  eccentricity.     Then 

e 

(33)  becomes 

iv 


This  is  Newton's  law  which  was  derived  from  the  same  conditions 
in  Art.  55. 

VTII.     PROBLEMS. 

C  C 

1.  Find  the  vis  viva  integral  when  /  =  -^,/  =  cr,  /  =  —  . 

2.  Suppose  that  in  Art.  53  the  particle  is  projected  orthogonally  from 
the  z-axis;  find  the  equations  corresponding  to  (19)  and  (20).     Suppose  still 
further  that  k  =  1,  XQ  =  1;  find  the  initial  velocity  such  that  the  eccentricity 
of  the  ellipse  may  be  1/2. 

or 
Am. 


3.  Find  the  central  force  as  a  function  of  the  distance  under  which  a 
particle  may  describe  the  spiral  r  =  —  ;  the  spiral  r  =  e0. 

h*  2h* 

Ans.    /  =  ^-  ,        /  =  —j-  . 

4.  Find  the  central  force  as  a  function  of  the  distance  under  which  a 
particle  may  describe  the  lemniscate  r2  =  a2  cos  20. 

Ans-    f  = 


5.  Find  the  central  force  as  a  function  of  the  distance  under  which  a 
particle  may  describe  the  cardioid  r  =  a(l  +  cos  6). 

Ans.    f  = 


90  ORBIT    FOR    FORCE  VARYING  AS   DISTANCE.  [61 

6.  Suppose  the  particle  describes  an  ellipse  with  the  origin  in  its  interior, 
at  a  distance  n  from  the  x-axis  and  m  from  the  ?/-axis.  (a)  Show  that  two  of 
the  laws  of  force  are 

r  ,  =  W (oc)i 

r2  [2mn  sin  0  cos  0  +  (a  -  c  -  n2  +  m2)  cos2  0  +  c  -  m2]* ' 


L         [ac  —  am2  —  en2  —  cny  —  amx]3 ' 

where  a  and  c  have  the  same  meaning  as  in  (26),  and  where  the  polar  axis 
is  parallel  to  the  major  axis  of  the  ellipse.  (6)  If  the  origin  is  between  the 
center  and  the  focus  show  that  the  force  at  unit  distance  is  a  maximum  for 

0  =  0  and  is  a  minimum  for  0  =  —  ;  that  if  the  origin  is  between  a  focus  and 

JB 

the  nearest  apse  the  maximum  is  for  0  =  —  and  the  minimum  for  0  =  0;  and 
that  if  the  origin  is  on  the  minor  axis  the  maximum  is  for  0  =  0,  and  the 
minimum  for  0  =  -~  . 

7.  Interpret  equation  (29)  geometrically. 
Hint.     C  sin  20  +  D  cos  20  +  H  =  (dx  +  /y)2  +  g(fl?9  +  Cy* 


The  numerator  of  this  expression  set  equal  to  zero  is  the  equation  of  the 
tangents  (real  or  imaginary)  from  the  origin  to  the  conic.  (Salmon's  Conic 
Sections,  Art.  92.) 

8.  Find  expressions  for  the  central  force  when  the  orbit  is  an  ellipse 
with  the  origin  at  an  end  of  the  major  and  minor  axes  respectively.     Show 

k2 
that  they  reduce  to  -^  when  the  ellipse  becomes  a  circle. 


arz     cos3  0  ' 


cr2      sin30' 


DETERMINATION  OF  THE  ORBIT  FROM  THE  LAW  OF  FORCE. 

61.  Force  Varying  as  the  Distance.  The  problem  of  finding 
the  orbit  when  the  law  of  force  is  given  is  generally  more  difficult 
than  the  converse,  since  it  involves  the  integration  of  (25).  The 
method  of  integration  varies  with  the  different  laws  of  force,  and 
the  character  of  the  integrals  depends  upon  the  initial  conditions. 
The  process  will  be  illustrated  first  in  the  case  in  which  the  force 
varies  as  the  distance,  a  problem  treated  by  another  method  in 
Art.  53. 

If  /  =  k2r,  equation  (25)  becomes 


61]  ORBIT  FOR  FORCE   VARYING  AS   DISTANCE.  91 


or 

d?u  =  k2  I 
d62  ~  h*  u 

The  first  integral  of  this  equation  is 

(du\2=    _Wl_ 
\dd)  h2u2 

whence 

(34)  de  =  ±udu 


Let 


_  _     A  2 

4       Jj?      4  * 


The  constant  A2  must  be  positive  in  order  that  -TT  may  be  real,  as 

du 

it  is  if  the  particle  is  started  with  real  initial  conditions. 
If  the  upper  sign  is  used,  equation  (34)  becomes 

(35)  2dd  = 


It  is  easily  verified  that  the  same  equation  (36)  would  be  reached, 
when  the  initial  conditions  are  substituted,  if  the  lower  sign  were 
used.  The  integral  of  (35)  is 


or 

z  =  A  cos  2(0  +  c2). 

On  going  back  to  the  variable  r,  this  equation  becomes 

2 
=  ci  -  2A  cos  2(0  +  c2)  ' 

This  is  the  polar  equation  of  an  ellipse  with  the  origin  at  the  center. 
Hence,  a  particle  moving  subject  to  an  attractive  force  varying 
directly  as  the  distance  describes  an  ellipse  with  the  origin  at  the 
center.  The  only  exceptions  are  when  the  particle  passes  through 
the  origin,  and  when  it  describes  a  circle.  In  the  first  h  =  0, 
and  equation  (25)  ceases  to  be  valid;  in  the  second,  c\  has  such  a 
value  that  it  satisfies  the  equation 


92  FORCE   VARYING   INVERSELY  AS   SQUARE  [62 

(du\*  k2     1 

U)o=    -/^-WO+C1  =  °' 

and  the  equation  of  the  orbit  becomes  u  =  UQ.  In  this  case 
equation  (34)  fails. 

62.  Force  Varying  Inversely  as  the  Square  of  the  Distance. 

Suppose  a  particle  moves  under  the  influence  of  a  central  attraction 
the  intensity  of  which  varies  inversely  as  the  square  of  the  distance  ; 
it  is  required  to  determine  its  orbit  when  it  is  projected  in  any 
manner.  Equation  (25)  is  in  this  case 

,Q7x  d2u      k2 

(37)  de2  =  h2  ~  u' 

This  equation  can  be  written  in  the  form 

d?u  .  k2 


This  is  a  linear  non-homogeneous  differential  equation  and  can 
be  integrated  by  the  method  of  variation  of  parameters,  which 
was  explained  in  Art.  37.  When  its  right  member  is  neglected 
the  general  solution  is 

u  =  Ci  cos  8  H-  c2  sin  0. 

k2 

It  is  clear  that  if  -^  is  added  to  this  value  of  u  the  differential 
h* 

equation  will  be  identically  satisfied.  Consequently  the  general 
solution  of  (37),  which  is  the  same  as  that  found  by  the  variation 
of  parameters,  is 

k2 

u  —  T5  +  ci  cos  0  +  C2  sm  ^' 
h2 

On  taking  the  reciprocal  of  this  equation,  it  is  found  that 

1 


r  = 


k2 

-^  +  GI  cos  6  +  C2  sin  0 

h2 


Now  let  Ci  =  A  cos  0o,  c2  =  A  sin  00,  where  A  and  00  are  constants. 
It  is  clear  that  A  can  always  be  taken  positive  and  equal  to 
Vci2  +  c22  and  a  real  00  can  be  determined  so  that  these  equations 
will  be  satisfied  whatever  real  values  Ci  and  C2  may  have.  Then 
the  equation  for  the  orbit  becomes 


(38) 


5  +  A  cos  (0  - 


63]  AND   INVERSELY   AS   FIFTH   POWER   OF   DISTANCE.  93 

This  is  the  polar  equation  of  a  conic  with  the  origin  at  one  of  the 
foci. 

From  this  investigation  and  that  of  Art.  55  it  follows  that  if  the 
orbit  is  a  conic  section  with  the  origin  at  one  of  the  foci,  and  the 
force  depends  on  the  distance  alone,  then  the  body  moves  subject 
to  a  central  force  varying  inversely  as  the  square  of  the  distance; 
and  conversely,  if  the  force  varies  inversely  as  the  square  of  the 
distance,  then  the  body  will  describe  a  conic  section  with  the 
origin  at  one  of  the  foci. 

Let  p  represent  the  parameter  of  the  conic  and  e  its  eccentricity. 
Then,  comparing  (38)  with  the  ordinary  polar  equation  of  the 

f) 

conic,  r  =  ^  -  ,  it  is  found  that 

1  +  e  cos  0 


(39) 


h* 
P  =17* 


and  0o  is  the  angle  between  the  polar  axis  and  the  end  of  the 
major  axis  directed  to  the  farthest  apse.  The  constants  h2  and  A 
are  determined  by  the  initial  conditions,  and  they  in  turn  define 
p  and  e  by  (39).  If  e  <  1,  the  conic  is  an  ellipse;  if  e  =  1,  the 
conic  is  a  parabola;  lie  >  1,  the  conic  is  a  hyperbola;  and  if  e  =  0, 
the  conic  is  a  circle. 

63.  Force  Varying  Inversely  as  the  Fifth  Power  of  the  Distance. 

k2 

In  this  case  /  =  -g  ,  and  (25)  becomes 


(40) 

u 
On  solving  for  -r—  •  and  integrating,  it  is  found  that 


Therefore 

(42)  de  -  du 


The  right  member  of  this  equation  cannot  in  general  be  integrated 
in  terms  of  the  elementary  functions,  but  it  can  be  transformed 
into  an  elliptic  integral  of  the  first  kind.  Then  u,  and  conse- 
quently r,  is  expressible  in  terms  of  0  by  elliptic  functions,  and  the 


94  FORCE   VARYING   INVERSELY  AS  [63 

orbits  in  general  either  wind  into  the  origin  or  pass  out  to  infinity, 
their  particular  character  depending  upon  the  initial  conditions. 

There  are  certain  special  cases  which  are  integrable  in  terms  of 
elementary  functions. 

(a)  If  such  a  constant  value  of  u  is  taken  that  it  fulfills  (41) 
when  its  right  member  is  set  equal  to  zero,  then  r  is  a  constant 
and  the  orbit  is  a  circle  with  the  origin  at  the  center.  It  is  easily 
seen  that  a  similar  special  case  will  occur  for  a  central  force  vary- 
ing as  any  power  of  the  distance. 

(6)  Another  special  case  is  that  in  which  the  initial  conditions 
are  such  that  cx  =h  0  and  the  right  member  of  (41)  is  a  perfect 

h2 
square.     That  is,  c\  =     -.     Then  equation  (41)  becomes 


_  .      _   u>__ 

u  ~A  • 


The  integral  of  this  equation  is 

,  1    +    A*U 

lo*l-A*u 
whence 


_ 

where  coth  —^-  (  =t  6  —  c2)  is  the  hyperbolic  cotangent  of 

iV2(±  0-c2). 

(c)  If  the  initial  conditions  are  such  that  c\  =  0,  equation  (41) 
gives 


*  a  .          *' 


the  integral  of  which  is 


On  taking  the  cosines  of  both  members  and  solving  for  r,  the  polar 
equation  of  the  orbit  is  found  to  be 

k 
(44)  r  =  -j=-  cos  (c2  =F  0)  , 

which  is  the  equation  of  a  circle  with  the  origin  on  the  circum- 
ference. 


63]  THE   FIFTH   POWER   OF    THE   DISTANCE.  95 

(d)  If  none  of  these  conditions  is  fulfilled  the  right  member  of 
(41)  is  a  biquadratic,  and  equation  (42)  can  be  written  in  the  form 

,A  r\  7fl  Cdu 

(45)  =±=  d&  =  7 


where  C,  a2,  and  /32  are  constants  which  depend  upon  the  coefficients 
of  (41)  in  a  simple  manner.  Equation  (45)  leads  to  an  elliptic 
integral  which  expresses  0  in  terms  of  u.  On  taking  the  inverse 
functions  and  the  reciprocals,  r  is  expressed  as  an  elliptic  function 
of  0.  The  curves  are  spirals  of  which  the  circle  through  the  origin, 
and  the  one  around  the  origin  as  center,  are  limiting  cases. 

If  the  curve  is  a  circle  through  the  origin  the  force  varies  in- 
versely as  the  fifth  power  of  the  distance  (Art.  56);  but  if  the 
force  varies  inversely  as  the  fifth  power  of  the  distance,  the  orbits 
which  the  particle  will  describe  are  curves  of  which  the  circle  is  a 
particular  limiting  case.  On  the  other  hand,  if  the  orbit  is  a 
conic  with  the  origin  at  the  center  or  at  one  of  the  foci,  the  force 
varies  directly  as  the  distance,  or  inversely  as  the  square  of  the 
distance ;  and  conversely,  if  the  force  varies  directly  as  the  distance, 
or  inversely  as  the  square  of  the  distance,  the  orbits  are  always 
conies  with  the  origin  at  the  center,  or  at  one  of  the  foci  respectively 
[Arts.  53,  55,  56  (6)].  A  complete  investigation  is  necessary  for 
every  law  to  show  this  converse  relationship. 

IX.     PROBLEMS. 

1.  Discuss  the  motion  of  the  particle  by  the  general  method  for  linear 
equations  when  the  force  varies  inversely  as  the  cube  of  the  distance.     Trace 
the  curves  in  the  various  special  cases. 

2.  Express  C,  a2,  and  /92  of  equation  (45)  in  terms  of  the  initial  conditions. 
For  original  projections  at  right  angles  to  the  radius  vector  investigate  all  the 
possible  cases,  reducing  the  integrals  to  the  normal  form,  and  expressing  r  as 
elliptic  functions  of  6.     Draw  the  curves  in  each  case. 

3.  Suppose  the  law  of  force  is  that  given  in  (29);  i.  e. 

M  M 


•e  — 


rz(C  sin  26  +  D  cos  26  + 


Integrate  the  differential  equation  of  the  orbit,  (25),  by  the  method  of  vari- 
ation of  parameters,  and  show  that  the  general  solution  has  the  form 

1  

-  =  Ci  cos  6  +  Cz  sin  d  +  "V0(0), 

where  c\  and  Cn  are  constants  of  integration.     Show  that  the  curve  is  a  conic. 


96  PROBLEMS. 


4.  When  the  force  is  /  =  -^  +  -^  show  that,  if  v  <  A2,  the  general  equa- 
tion of  the  orbit  described  has  the  form 

a 


I  -  e  cos(^)  ' 

where  a,  e,  and  k  are  the  constants  depending  upon  the  initial  conditions  and 
fj.  and  v.  Observe  that  this  may  be  regarded  as  being  a  conic  section  whose 
major  axis  revolves  around  the  focus  with  the  mean  angular  velocity 

n  =  (l-fc)Y' 
where  T  is  the  period  of  revolution. 

5.  In  the  case  of  a  central  force  the  motion  along  the  radius   vector  is 
denned  by  the  equation 

^=_/  +  ^! 

dP  J  ^  r3  ' 

Discuss  the  integration  of  this  equation  when 


6.  Suppose  the  law  of  force  is  that  given  by  (30)  ;  i.  e., 

N 


r2  (  -  —  A  sin  e  —  B  cos  0  Y 


Substitute  in  (25)  and  derive  the  general  equation  of  the  orbit  described. 
Hint.     Let  u  =  v  +  Asm&-\-B  cos  6',  then  (25)  becomes 


Ans.    -  =  A  sin  6  +  B  cos  6  +  Vci  cos2  6  +  c2  sin  20  +  c3  sin2  9, 
r 

which  is  the  equation  of  a  conic  section. 

7.  Suppose  the  law  of  force  is 

_  ci  +  c2  cos  26 

/    •  r2 

show  that,  for  all  initial  projections,  the  orbit  is  an  algebraic  curve  of  the 
fourth  degree  unless  c2  =  0,  when  it  reduces  to  a  conic. 


HISTORICAL   SKETCH.  97 


HISTORICAL   SKETCH  AND  BIBLIOGRAPHY. 

The  subject  of  central  forces  was  first  discussed  by  Newton.  In  Sections 
ii.  and  in.  of  the  First  Book  of  the  Principia  he  gave  a  splendid  geometrical 
treatment  of  the  subject,  and  arrived  at  some  very  general  theorems.  These 
portions  of  the  Principia  especially  deserve  careful  study. 

All  the  simpler  cases  were  worked  out  in  the  eighteenth  century  by  analyti- 
cal methods.  A  few  examples  are  given  in  detail  in  Legendre's  Traite  des 
Fonctions  Elliptiques.  An  exposition  of  principles  and  a  list  of  examples 
are  given  in  nearly  every  work  on  analytical  mechanics;  among  the  best  of 
these  treatments  are  the  Fifth  Chapter  in  Tait  and  Steele's  Dynamics  of  a 
Particle,  and  the  Tenth  Chapter,  vol.  i.,  of  AppelFs  Mecanique  Rationelle. 
Stader's  memoir,  vol.  XLVI.,  Journal  fur  Mathematik,  treats  the  subject  in 
great  detail.  The  special  problem  where  the  force  varies  inversely  as  the 
fifth  power  of  the  distance  has  been  given  a  complete  and  elegant  treatment 
by  MacMillan  in  The  American  Journal  of  Mathematics,  vol.  xxx,  pp.  282-306. 

The  problem  of  finding  the  general  expression  for  the  possible  laws  of  force 
operating  in  the  binary  star  systems  was  proposed  by  M.  Bertrand  in  vol. 
LXXXIV.  of  the  Comptes  Rendus,  and  was  immediately  solved  by  MM.  Darboux 
and  Halphen,  and  published  in  the  same  volume.  The  treatment  given  above 
in  the  text  is  similar  to  that  given  by  M.  Darboux,  which  has  also  been  repro- 
duced in  a  note  at  the  end  of  the  Mecanique  of  M.  Despeyrous.  The  method 
of  M.  Halphen  is  given  in  Tisserand's  Me  anique  Celeste,  vol.  i.,  p.  36,  and  in 
Appell's  Mecanique  Rationelle,  vol.  i.,  p.  372.  It  seems  to  have  been  generally 
overlooked  that  Newton  had  reated  the  same  problem  in  the  Principia, 
Book  i.,  Scholium  to  Proposition  xvn.  This  was  reproduced  and  shown  to 
be  equivalent  to  the  work  of  MM.  Darboux  and  Halphen  by  Professo:  Glaisher 
in  the  Monthly  Notices  f  R.A.S.,  vol.  xxxix. 

M.  Bertrand  has  shown  (Comptes  Rendus,  vol.  LXXVII.)  that  the  only  laws 
of  central  force  under  the  action  of  which  a  particle  will  describe  a  conic 

fc2 
section  for  all  initial  conditions  are/  =  =*=  -^  and/  =  ±  k*r.    M.  Koenigs  has 

proved  (Bulletin  de  la  Societe  Mathematique,  vol.  xvn.)  that  the  only  laws  of 
central  force  depending  upon  the  distance  alone,  for  which  the  curves  de- 
ft2 
scribed  are  algebraic  for  all  initial  conditions    are  /  =  =*=  -5  and  /  =  =*=  &2r. 

Griffin  has  shown  (American  Journal  of  Mathematics,  vol.  xxxi.)  that  the 
only  law,  where  the  force  is  a  function  of  the  distance  alone,  where  it  does  not 
vanish  at  the  center  of  force,  and  where  it  is  real  throughout  the  plane,  giving 
an  elliptical  orbit  is  the  Newtonian  law. 


CHAPTER  IV. 

THE  POTENTIAL   AND    ATTRACTIONS    OF   BODIES. 

64.  THE  previous  chapters  have  been  concerned  with  problems 
in  which  the  law  of  force  was  given,  or  with  the  discovery  of  the 
law  of  force  when  the  orbits  were  given.     All  the  investigations 
were  made  as  though  the  masses  were  mere  points  instead  of  being 
of  finite  size.     When  forces  exist  between  every  two  particles  of 
all  the  masses  involved,  bodies  of  finite  size  cannot  be  assumed  to 
attract  one  another  according  to  the  same  laws.     Hence  it  is  neces- 
sary to  take  up  the  problem  of  determining  the  way  in  which 
finite  bodies  of  different  shapes  attract  one  another. 

It  follows  from  Kepler's  laws  and  the  principles  of  central  forces 
that,  if  the  planets  are  regarded  as  being  of  infinitesimal  dimen- 
sions compared  to  their  distances  from  the  sun,  they  move  under 
the  influence  of  forces  which  are  directed  toward  the  center  of  the 
sun  and  which  vary  inversely  as  the  squares  of  their  distances 
from  it.  This  suggests  the  idea  that  the  law  of  inverse  squares 
may  account  for  the  motions  still  more  exactly  if  the  bodies  are 
regarded  as  being  of  finite  size,  with  every  particle  attracting  every 
other  particle  in  the  system.  The  appropriate  investigation  shows 
that  this  is  true. 

This  chapter  will  be  devoted  to  an  exposition  of  general  methods 
of  finding  the  attractions  of  bodies  of  any  shape  on  unit  particles 
in  any  position,  exterior  or  interior,  when  the  forces  vary  inversely 
as  the  squares  of  the  distances.  The  astronomical  applications 
will  be  to  the  attractions  of  spheres  and  oblate  spheroids,  to  the 
variations  in  the  surface  gravity  of  the  planets,  and  to  the  per- 
turbations of  the  motions  of  the  satellites  due  to  the  oblateness  of 
the  planets. 

65.  Solid  Angles.     If  a  straight  line  constantly  passing  through 
a  fixed  point  is  moved  until  it  retakes  its  original  position,  it  gener- 
ates a  conical  surface  of  two  sheets  whose  vertices  are  at  the  given 
point.     The  area  which  one  end  of  this  double  cone  cuts  out  of  the 
surface  of  the  unit  sphere  whose  center  is  at  the  given  point  is 
called  the  solid  angle  of  the  cone;  or,  the  area  cut  out  of  any  con- 

98 


65]  SOLID  ANGLES.  99 

centric  sphere  divided  by  the  square  of  its  radius  measures  the 
solid  angle. 

Since  the  area  of  a  spherical  surface  equals  the  product  of  4?r 
and  the  square  of  its  radius,  it  follows  that  the  sum  of  all  the  solid 
angles  about  a  point  is  4vr.  The  sum  of  the  solid  angles  of  one-half 
of  all  the  double  cones  which  can  be  constructed  about  a  point 
without  intersecting  one  another  is  2?r. 

The  volume  contained  within  an  infinitesimal  cone  whose  solid 
angle  is  co  and  between  two  spherical  surfaces  whose  centers  are 
at  the  vertex  of  the  cone,  approaches  as  a  limit,  as  the  surfaces 
approach  each  other,  the  product  of  the  solid  angle,  the  square  of 
the  distance  of  the  spherical  surfaces  from  the  vertex,  and  the 
distance  between  them.  If  the  centers  of  the  spherical  surfaces 
are  at  a  point  not  in  the  axis  of  the  cone,  the  Volume  approaches 
as  a  limit  the  product  of  the  solid  angle,  the  square  of  the  distance 


Fig.  10. 

from  the  vertex,  the  distance  between  the  spherical  surfaces,  and 
the  reciprocal  of  the  cosine  of  the  angle  between  the  axis  of  the 
cone  and  the  radius  from  the  center  of  the  sphere;  or,  it  is  the 
product  of  the  solid  angle,  the  square  of  the  distance  from  the 
vertex,  and  the  intercept  on  the  cone  between  the  spherical 
surfaces.  Thus,  the  volume  of  abdc,  Fig.  10,  is  V  =  o>a02  •  ab. 
The  volume  of  a'b'd'c'  is 

T7,      coa'Q2  •  b'e'         -^      ,,, 

*     ~ fr\  tr\t\  =  uaO   -ab. 

cos  (Oa'O) 

Sometimes  it  will  be  convenient  to  use  one  of  these  expressions  and 
sometimes  the  other. 

66.  The  Attraction  of  a  Thin  Homogeneous  Spherical  Shell 
upon  a  Particle  in  its  Interior.  The  attractions  of  spheres  and 
other  simple  figures  were  treated  by  Newton  in  the  Principia, 


100  ATTRACTION   OF    ELLIPSOIDAL    SHELLS  [67 

Book  i.,  Section  12.     The  following  demonstration  is  essentially 
as  given  by  him. 

Consider  the  spherical  shell  contained  between  the  infinitely 
near  spherical  surfaces  S  and  S',  and  let  P  be  a  particle  of  unit 
mass  situated  within  it.  Construct  an  infinitesimal  cone  whose 


A' 

Fig.  11. 

solid  angle  is  w  with  its  vertex  at  P.  Let  a  be  the  density  of 
the  shell.  Then  the  mass  of  the  element  of  the  shell  at  A  is 
m  =  aABuAP  ;  likewise  the  mass  of  the  element  of  A'  is 
m'  —  vA'B'uA'P  .  The  attractions  of  m  and  m!  upon  P  are 
respectively 

k2mf 


a.  =  =3  ,         a    = 
AP 


Since  A'B'  =  AB,  a  =  WABuv  =  a'.  This  holds  for  every  infini- 
tesimal solid  angle  with  vertex  at  P;  therefore  a  thin  homogeneous 
spherical  shell  attracts  particles  within  it  equally  in  opposite  directions. 
This  holds  for  any  number  of  thin  spherical  shells  and,  therefore, 
for  shells  of  finite  thickness. 

67.  The  Attraction  of  a  Thin  Homogeneous  Ellipsoidal  Shell 
upon  a  Particle  in  its  Interior.  The  theorem  of  this  article  is 
given  in  the  Principia,  Book  i.,  Prop,  xci.,  Cor.  3. 

Let  a  homoeoid  be  defined  as  a  thin  shell  contained  between  two 
similar  surfaces  similarly  placed.  Thus,  an  elliptic  homoeoid  is  a 
thin  shell  contained  between  two  similar  ellipsoidal  surfaces  simi- 
larly placed. 

Consider  the  attraction  of  the  elliptic  homoeoid  whose  surfaces 
are  the  similar  ellipsoids  E  and  Ef  upon  the  interior  unit  particle  P. 
Construct  an  infinitesimal  cone  whose  solid  angle  is  co  with  vertex 


68] 


UPON   AN   INTERIOR   PARTICLE. 


101 


at  P.  The  masses  of  the_two_infinitesimal  elements  at  A  and  A' 
are  respectively  m  =  aABuAP  and  mr  =  aA'B'uA'P*.  The 

k2m  k^m' 

attractions  are  a  =  =5  and  a   =  -==5 .      Construct  a  diameter 
AP  A  P 

CC'  parallel  to  A  A'  in  the  elliptical  section  of  a  plane  throughjhe 
cone  and  the  center  of  the  ellipsoids,  and  draw  its  conjugate  DD' . 
They  are  conjugate  diameters  in  both  elliptical  sections,  E  and 
E'\  therefore  DD'  bisects  every  chord  parallel  to  CC' ,  and  hence 
AB  =  A'B'.  The  attractions  of  the  elements  at  A  and  A'  upon 


P  are  therefore  equal.  This  holds  for  every  infinitesimal  solid 
angle  whose  vertex  is  at  P;  therefore  the  attractions  of  a  thin  elliptic 
homoeoid  upon  an  interior  particle  are  equal  in  opposite  directions. 
This  holds  for  any  number  of  thin  shells  and,  therefore,  for 
shells  of  finite  thickness. 

68.  The  Attraction  of  a  Thin  Homogeneous  Spherical  Shell 
upon  an  Exterior  Particle.     Newton's   Method.     Let   AH  KB 

and  ahkb  be  two  equal  thin  spherical  shells  with  centers  at  0  and  o 


Fig.  13. 


respectively.  Let  two  unit  particles  be  placed  at  P  and  p,  unequal 
distances  from  the  centers  of  the  shells.  Draw  any  secants  from  p 
cutting  off  the  arcs  il  and  hk,  and  let  the  angle  kpl  approach  zero 
as  a  limit.  Draw  from  P  the  secants  PL  and  PK,  cutting  off  the 


102  ATTRACTION   OF   THIN    SPHERICAL   SHELLS  [68 

arcs  IL  and  HK  equal  respectively  to  il  and  hk.  Draw  oe  per- 
pendicular to  pi,  od  perpendicular  to  pk,  iq  perpendicular  to  pb, 
and  ir  perpendicular  to  pk.  Draw  the  corresponding  lines  in  the 
other  figure. 

Rotate  the  figures  around  the  diameters  PB  and  pb,  and  call 
the  masses  of  the  circular  rings  generated  by  HI  and  hi,  M  and  m 
respectively;  then 

(1)  HI  X  IQ  :  hi  X  iq  =  M  :  m. 

The  attractions  of  unit  masses  situated  at  /  and  i  are  respectively 
proportional  to  the  inverse  squares  of  PI  and  pi.  The  com- 
ponents of  these  attractions  in  the  directions  PO  and  po  are  the 

respective  attractions  multiplied  by  ~p  and  —.  respectively.     If 

£T J.  jUv 

A'  and  a'  represent  the  components  of  attraction  toward  0  and  o, 
then 

(21  A>.a>  -±-PQ.±n 

~  PI'  PI  V  Pi' 

Now  consider  the  attractions  of  the  rings  upon  P  and  p.  Their 
resultants  are  in  the  directions  of  0  and  o  respectively  because 
of  the  symmetry  of  the  figures  with  respect  to  the  lines  PO  and  po, 
and  they  are  respectively  M  and  m  times  those  of  the  unit  particles. 
Let  A  and  a  represent  the  attractions  of  M  and  m;  then 

M.  PQ  ^L  £2  =  HI  x  7#  ?E  hi  x  i(j  vf_ 

~PI*PI:^pi=        Pf    ~  PC*'      pf      po' 

In  order  to  reduce  the  right  member  of  (3)  consider  the  similar 
triangles  PIR  and  PFD  and  the  corresponding  triangles  in  the 
other  figure.  At  the  limit  as  the  angles  KPL  and  kpl  approach 
zero,  DF  :  df  =  1  because  the  secants  IL  and  HK  respectively 
equal  il  and  hk.  Therefore 

PI  :  PF  =  RI  :  DF, 

pf:pi  =  DF(=  df)  :ri, 
and  the  product  of  these  proportions  is 

(4)  PI  X  pf  :  PF  X  pi  =  RI  :  ri  =  HI  :  hi. 

From  the  similar  triangles  PIQ  and  POE,  it  follows  that 

PI  :  PO  =  IQ  :  OE, 
and  similarly 

po  :  pi  =  OE(  =  oe)  :  iq. 


69]  UPON    AN    EXTERIOR   PARTICLE.  103 

The  product  of  these  two  proportions  is 

(5)  PI  X  po  :  PO  X  pi  =  IQ  :  iq. 
The  product  of  (4)  and  (5)  is 

Pl'xpfXpo:  pi  X  PF  X  PO  =  HI  X  IQ  :  hi  X  iq. 
Consequently  equation  (3)  becomes 

(6)  A  :  a  =  po*  :  PO*. 

Therefore,  the  circular  rings  attract  the  exterior  particles  toward 
the  centers  of  the  shells  with  forces  which  are  inversely  propor- 
tional to  the  squares  of  the  respective  distances  of  the  particles 
from  these  centers.  In  a  similar  manner  the  same  can  be  proved 
for  the  rings  KL  and  kl. 

Now  let  the  lines  PK  and  pk  vary  from  coincidence  with  the 
diameters  PB  and  pb  to  tangency  with  the  spherical  shells.  The 
results  are  true  at  every  position  separately,  and  hence  for  all  at 
once.  Therefore,  the  resultants  of  the  attractions  of  thin  spherical 
shells  upon  exterior  particles  are  directed  toward  their  centers,  and 
the  intensities  of  the  forces  vary  inversely  as  the  squares  of  the  distances 
of  the  particles  from  the  centers. 

If  the  body  is  a  homogeneous  sphere,  or  is  made  up  of  homo- 
geneous spherical  layers,  the  theorem  holds  for  each  layer  sepa- 
rately, and  consequently  for  all  of  them  combined. 

69.  Comments  upon  Newton's  Method.  While  the  demon- 
stration above  is  given  in  the  language  of  Geometry,  it  really 
depends  upon  the  principles  which  are  fundamental  in  the  Calculus. 
Letting  the  angle  kpl  approach  zero  as  a  limit  is  equivalent  to 
taking  a  differential  element;  the  rotation  around  the  diameters  is 
equivalent  to  an  integration  with  respect  to  one  of  the  polar  angles; 
the  variation  of  the  line  pk  from  coincidence  with  the  diameter  to 
tangency  with  the  shell  is  equivalent  to  an  integration  with  respect 
to  the  other  polar  angle;  and  the  summation  of  the  infinitely  thin 
shells  to  form  a  solid  sphere  is  equivalent  to  an  integration  with 
respect  to  the  radius. 

Since  Newton's  method  gives  only  the  ratios  of  the  attraction  of 
equal  spherical  shells  at  different  distances,  it  does  not  give  the 
manner  in  which  the  attraction  depends  upon  the  masses  of  the 
finite  bodies.  This  is  of  scarcely  less  importance  than  a  knowl- 
edge of  the  manner  in  which  it  varies  with  the  distance. 

In  order  to  find  the  manner  in  which  the  attraction  depends  upon 
the  mass  of  the  attracting  body,  take  two  equally  dense  spherical 


104  THOMSON   AND   TAIT's  METHOD.  [70 

shells,  Si  and  $2,  internally  tangent  to  the  cone  C.  Let  POi  =  ai, 
P02  =  «2,  and  MI  and  M2  be  the  masses  of  Si  and  *S2  respectively. 
The  two  shells  attract  the  particle  P  equally;  for,  any  solid  angle 
which  includes  part  of  one  shell  also  includes  a  similar  part  of  the 
other.  The  masses  of  these  included  parts  are  as  the  squares  of 


Fig.  14. 


their  distances,  and  their  attractions  are  inversely  as  the  squares 
of  their  distances,  whence  the  equality  of  their  attractions  upon 
P.  Let  A  represent  the  common  attraction;  then  remove  Si  so 
that  its  center  is  also  at  02.  Let  A'  represent  the  intensity  of  the 
attraction  of  Si  in  the  new  position;  then,  by  the  theorem  of 
Art.  68, 

A!_  =  oi2  =  Mi 

A       a22      M2 ' 

Therefore,  the  two  shells  attract  a  particle  at  the  same  distance  with 
forces  directly  proportional  to  their  masses.  From  this  and  the 
previous  theorem,  it  follows  that  a  particle  exterior  to  a  sphere  which 
is  homogeneous  in  concentric  layers  is  attracted  toward  its  center 
with  a  force  which  is  directly  proportional  to  the  mass  of  the  sphere 
and  inversely  as  the  square  of  the  distance  from  its  center-,  or,  as 
though  the  mass  of  the  sphere  were  all  at  its  center. 

Since  the  heavenly  bodies  are  nearly  homogeneous  in  concentric 
spherical  layers  they  can  be  regarded  as  material  points  in  the  dis- 
cussion of  their  mutual  interactions  except  when  they  are  relatively 
near- each  other  as  in  the  case  of  the  planets  and  their  respective 
satellites. 

70.  The  Attraction  of  a  Thin  Homogeneous  Spherical  Shell 
upon  an  Exterior  Particle.  Thomson  and  Tait's  Method.  Let 
0  be  the  center  of  the  spherical  shell  whose  radius  is  a  and  whose 
thickness  is  Ac,  P  the  position  of  the  attracted  particle  and  PO  a 
line  from  the  attracted  particle  to  the  center  cutting  the  spherical 
surface  in  C.  Take  the  point  A  so  that  PO  :  OC  =  OC  :  OA,  and 
construct  the  infinitesimal  cone  whose  solid  angle  is  co  with  its 
vertex  at  A.  Let  a  be  the  density  of  the  shell.  Then  the  elements 
of  mass  at  B  and  B'  are  respectively 


70] 


m  = 


THOMSON   AND   TAIT*S  METHOD. 

-r^i        Aa  ,  -r-^-72         Aa 


105 


m'  =  ™AB' 


cos  (OBA)  >  cos  (OB'A) ' 

The  attractions  of  the  two  masses  upon  P  are  respectively 

Aa 


(7) 


a  = 


a'  =  k2(ra> 


*     cos  (OBA) ' 

AB'*  Aa 

&p* '  cos  (OB' A) 


Fig.  15. 

From  the  construction  of  A  it  follows  that 
PO  :  OB  =  OB  :  OA. 

Hence  the  triangles  FOB  and  BOA,  having  a  common  angle  in- 
cluded between  proportional  sides,  are  similar.     Therefore 


AB 


Similarly 


BP 

AB' 
B'P 


OB 
OP 


a 
OP' 


a 
OP' 


The  angle  OBA   equals  the  angle  OB'A.     Then  equations   (7) 

become 

i2  Aa 


(8) 


OP2 '  cos  (OBA) ' 

a2  Aa 

<5p  '  cos  (OBA)  ~~ 


The  angles  BPO  and  B'PO  are  respectively  equal  to  OBA  and 
OB' A',  therefore  they  are  equal  to  each  other.  The  resultant  of 
the  two  equal  attractions  a  and  a'  is  in  the  line  bisecting  the  angle 
between  them,  or  in  the  direction  of  0,  and  is  given  in  magnitude 
by  the  equation 


106  ATTRACTION   UPON    PARTICLE   IN   THIN   SHELL.  [71 

Afl  =  a  cos  (BPO)  +  a'  cos  (B'PO)  =  2a  cos  (OB  A). 
This  becomes,  as  a  consequence  of  (8), 


OP 

This  equation  is  true  for  every  solid  angle  with  vertex  at  A,  and 
consequently  for  their  sum.  Therefore  the  attraction  of  the  whole 
spherical  shell  upon  the  exterior  particle  is,  on  summing  with 
respect  to  o>, 

D        .   I2   a2Aa      k2M 

R  =  47rA>2(7  ==-  =  ==-  : 
OP        OP 

or,  the  attraction  varies  directly  as  the  mass  of  the  shell  and  in- 
versely as  the  square  of  the  distance  of  the  particle  from  its  center. 

71.  The  Attraction  upon  a  Particle  in  a  Homogeneous  Spherical 
Shell.  In  Arts.  66-69  the  attractions  of  a  thin  homogeneous 
spherical  shell  upon  an  interior  and  an  exterior  particle,  respec- 
tively, have  been  discussed;  the  problem  is  now  completed  by 
treating  the  case  where  the  attracted  particle  is  a  part  of  the  shell 
itself. 

Let  0  be  the  center  of  the  spherical  shell  of  thickness  Aa, 
and  P  the  position  of  the  attracted  particle.  Construct  a  cone 
whose  solid  angle  is  co  with  its  vertex  at  P.  Let  a-  be  the  den- 


Fig.  16. 

sity  of  the  shell;  then  the  mass  of  the  section  cut  out  at  A  by 

the  cone  is   o-coAP  -      fr.  .  n.  .     The  attraction  of  the  element 
cos  (OAP) 

along  AP  is  a  =  ArW  ==5 (r.A  m  .     The  resultant  attraction 

AP  cos  (OAP) 

of  the  shell  is  in  the  direction  PO  since  the  mass  is  symmetrically 
situated  with  respect  to  this  line.  The  component  in  the  direction 
POis 

AR  =  a  cos  (APO)  =  a  cos  (OAP)  = 


PROBLEMS.  107 

The  attraction  of  the  whole  shell  is 
R  = 


It  follows  from  this  equation  and  the  results  obtained  in  Arts. 
66  and  69  that  the  attraction  on  an  interior  particle  infinitely  near 


the  shell  is  zero,  on  a  particle  in  the  shell,  --  ,  and  on  an  exterior 


particle  infinitely  near  the  shell,  —  5-  .*     The  discontinuity  in  the 

attraction  is  due  to  the  fact  that  the  mass  of  any  finite  area  of  the 
shell  is  assumed  to  be  finite  although  it  is  supposed  to  be  infinitely 
thin.  There  is  no  such  discontinuity  at  the  surface  of  a  solid  sphere 
because  an  infinitely  thin  shell  taken  from  it  has  only  an  infini- 
tesimal mass. 

X.     PROBLEMS. 

1.  Suppose  any  two  similar  bodies  are  similarly  placed  in  perspective. 
Show  that  a  particle  at  their  center  of  perspectivity  is  attracted  inversely  as 
their  linear  dimensions  if  they  are  thin  rods  of  equal  density;  equally,  if  they 
are  thin  shells  of  equal  density;  and  directly  as  their  linear  dimensions  if 
they  are  solids  of  equal  density.     Consider  a  nebula  which  is  apparently  as 
large  as  the  sun.     Suppose  its  distance  is  one  million  times  that  of  the  sun 
and  that  its  density  is  one  millionth  that  of  the  sun.     Compare  its  attraction 
for  the  earth  with  that  of  the  sun. 

2.  Prove  that  the  attractions  of  two  homogeneous  spheres  of  equal  density 
for  particles  upon  their  surfaces  are  to  each  other  as  their  radii. 

3.  Prove  that  the  attraction  of  a  homogeneous  sphere  upon  a  particle  in 
its  interior  varies  directly  as  the  distance  of  the  particle  from  the  center. 

4.  Prove  that  all  the  frustums  of  equal  height  of  any  homogeneous  cone 
attract  a  particle  at  its  vertex  equally. 

5.  Find  the  law  of  density  such  that  the  attraction  of  a  sphere  for  a  particle 
upon  its  surface  shall  be  independent  of  the  size  of  the  sphere. 

6.  Prove.  that  the  attraction  of  a  uniform  thin  rod,  bent  in  the  form  of 
an  arc  of  a  circle,  upon  a  particle  at  the  center  of  the  circle  is  the  same  as 
that  which  the  mass  of  a  similar  rod  equal  to  the  chord  joining  the  extremities 
would  exert  if  it  were  concentrated  at  the  middle  point  of  the  arc. 

7.  Prove  that  the  attraction  of  a  thin  uniform  straight  rod  on  an  exterior 
particle  is  the  same  in  magnitude  and  direction  as  that  of  a  circular  arc  of  the 
same  density,  with  its  center  at  the  particle  and  subtending  the  same  angle 
as  the  rod,  and  which  is  tangent  to  the  rod. 

*  See  note  on  the  attraction  of  spherical  shells,  Lagrange,  Collected  Works, 
vol.  vii.,  p.  591. 


108 


EQUATIONS  FOR  COMPONENTS  OF  ATTRACTION. 


[72 


8.  Prove  that  if  straight  uniform  rods  form  a  polygon  all  of  whose  sides 
are  tangent  to  a  circle,  a  particle  at  the  center  of  the  circle  is  attracted  equally 
in  opposite  directions  by  the  rods. 

9.  Prove  that  two  spheres,  homogeneous  in  concentric  spherical  layers, 
attract  each  other  as  though  their  masses  were  all  at  their  respective  centers. 

72.  The  General  Equations  for  the  Components  of  Attraction 
and  for  the  Potential  when  the  Attracted  Particle  is  not  a  Part 
of  the  Attracting  Mass.  The  geometrical  methods  of  the  pre- 
ceding articles  are  special,  being  efficient  only  in  the  particular 
cases  to  which  they  are  applied;  the  analytical  methods  which 
follow  are  characterized  by  their  uniformity  and  generality,  and 
illustrate  again  the  advantages  of  processes  of  this  nature. 

Consider  the  attraction  of  the  finite  mass  M  whose  density  is  a 
upon  the  unit  particle  P,  which  is  not  a  part  of  it.  That  is,  P  is 
exterior  to  M  or  within  some  cavity  in  it.  Let  the  coordinates 
of  P  be  x,  y,  z.  Let  the  coordinates  of  any  element  of  mass  dm 


Fig.  17. 

be  £,  y,  f  ,  and  the  distance  from  dm  to  P  be  p.    Then  the  com- 
ponents of  attraction  parallel  to  the  coordinate  axes  are  respectively 


(9)    1 


where 


--»f    **. 

J(M)     p2 


X 

Y  =  - 

Z  =  - 


M)  p 


p3 


dm, 


72] 


EQUATIONS   FOR   COMPONENTS   OF   ATTRACTION. 


109 


dm  =  <rd£diidt, 

P2  =  (x  -  £)2  +  (y  -  r;)2  +  (z  - 


The  integral  sign  J      signifies  that  the  integral  must  be  extended 

over  the  whole  mass  M.  Then,  if  a  is  a  finite  continuous  function 
of  the  coordinates,  as  will  always  be  the  case  in  what  follows,  X, 
Y,  and  Z  are  finite  definite  quantities.  In  practice  dm  is  expressed 
in  terms  of  <r  and  the  ordinary  rectangular  or  polar  coordinates, 
and  X,  Y,  and  Z  are  found  by  triple  integrations. 

The  three  integrals  (9)  can  be  made  to  depend  upon  a  single 
integral  in  a  very  simple  manner.     Let 

dm 


do) 


-L 


V  is  called  the  potential  function,  the  term  having  been  introduced 
by  Green  in  1828.  It  is  a  function  of  x,  y,  and  z  and  will  be 
spoken  of  as  the  potential  of  M  upon  P  at  the  point  (x,  y,  z). 

Since  P  is  not  a  part  of  the  mass  M,  p  does  not  vanish  in  the 
region  of  integration.  The  limits  of  the  integral  are  independent 
of  the  position  of  the  attracted  particle;  therefore  the  function 
under  the  integral  sign  can  be  differentiated  with  respect  to 
x,  y,  z  which  are  treated  as  constants  in  computing  the  definite 
integrals.  The  partial  derivatives  of  V  with  respect  to  x,  y 
and  z  are 


dx 


dy 

dv  =      r 

dz  ~      ~  J(A 


dm, 
dm. 


(M)          p 

On  comparing  these  equations  with  (9),  it  is  found  that 


(11) 


110  EQUATIONS   FOR    COMPONENTS   OF   ATTRACTION  [73 

Therefore,  in  the  case  in  which  P  is  not  a  part  of  M,  the  solution 
of  the  problem  of  finding  the  components  of  attraction  depends 
upon  the  computation  of  the  single  function  V. 

73.  Case  where  the  Attracted  Particle  is  a  Part  of  the  Attracting 
Mass.  It  will  now  be  proved  that  the  components  of  attraction 
and  the  potential  have  finite,  definite  values  when  the  particle  is 
a  part  of  the  attracting  mass,  and  that  equations  (11)  also  hold  in 
this  case. 

In  order  to  show  first  that  X,  Y,  Z,  and  V  have  finite,  determi- 
nate values  in  this  case,  let  dm  and  its  position  be  expressed  in 
polar  coordinates  with  the  origin  at  the  attracted  particle  P.  The 
equations  expressing  the  rectangular  coordinates  in  terms  of  the 
polar  with  the  origin  at  P  are 

—  x  =  p  cos  (p  cos  6, 

—  y  =  P  cos  (p  sin  0, 
-  z  =  p  sin  (p, 

dm  =  ap2  cos  (pd<pdddp. 

Then  the  expressions  for  the  components  of  attraction  and  the 
potential  become 

'  X  =  -  Wfffo-  cos2  (p  cosed<pd6dp, 
Y  =  -  k2fffa  cos2  <p  smed<pd6dp, 
Z  =  —  k2  fff  a  sin  <p  cos  (p  d<p  dd  dp, 
V  =  +  ///  ap  cos  <p  d(p  dd  dp, 

where  the  limits  are  to  be  so  determined  that  the  integration 
shall  be  extended  throughout  the  whole  body  M.  The  integrands 
are  all  finite  for  all  points  in  M,  and  therefore  the  integrals  have 
finite,  determinate  values. 

The  simplest  method  of  proving  that  equations  (11)  hold  when 
P  is  in  the  attracting  mass  M  is  to  start  from  the  definition  of  the 
derivative  of  V  with  respect  to  x.  By  definition 

ay  =  lim  v  -  v 

dx       A*=o      Ax 

where  V  is  the  potential  at  the  point  P1  whose  coordinates  are 
(x  -f  Ax,  y,  z).  Construct  a  small  sphere  of  radius  e  enclosing 
both  P  and  P'.  Let  the  mass  contained  within  the  sphere  c  be 


73] 


WHEN   PARTICLE   IS   PART   OF   ATTRACTING   MASS. 


Ill 


represented  by  M i  and  that  outside  of  it  by  7kf2.  Let  the  corre- 
sponding parts  of  the  components  of  attraction  and  the  potential 
be  distinguished  by  the  subscripts  1  and  2.  Then 


Fig.  18. 


(12) 


because  all  of  these  quantities  are  uniquely  defined.     Moreover, 
it  follows  from  Art.  72  that 


X*  =  k* 


dV, 


=  k< 


<972 


d7s 


dx  '  dy  '  dz  ' 

Now  consider  the  derivative  of  V  with  respect  to  x.     It  becomes 


(13) 


dV      .. 
—  =  hm 
dx       A*=o 


i  ,   ,. 
+  hm 


TV  -  72 


Let  the  distance  from  P  to  dm  be  p,  and  from  Pf  to  dm  be  pf. 
Then 


Ax 


_ 

pf      p  I  Ax  * 


It  follows  from  the  triangle  P  dm  P'  that  |  Ax  |  ^  |  p'  -  p  |, 
where  the  vertical  lines  on  a  quantity  indicate  that  its  numerical 
value  is  taken.  Hence  it  follows  that 


Therefore 


Ax 


=PP'=2 

<l  f   dm,,\C    dm 
-2  JIM  P2  +2j(j/,)/>'2  ' 


112  EQUATIONS  FOR   COMPONENTS   OF   ATTRACTION.  [73 

When  dm  is  expressed  in  polar  coordinates  this  inequality  becomes 


Ax 


i  ri  rZir  fp 

^  -  I  I    a  cos  <p  dtp  dd  dp 

*  J_  E  J0     Jo 


cos 


Let  o-o  be  the  maximum  value  of  <r  in  e.     The  result  of  integrating 
with  respect  to  p  and  p'  is 


TV  - 


P  cos 


cos 


Since  P  and  P'  are  in  the  sphere  e  the  distances  p  and  p'  cannot 
exceed  2c.     Then 


and 


IT 
/^T        /•2T 

I        I        cos  <p'd<p'dd'  =  STTO-Q  e, 


lim 

Aa:=0 


Vlf-Vl 


ft 

07T(7o  €. 


It  follows  from  this  inequality  and  (13)  that 


fc2        - 
ox 


<  /c2 
dx 


Now  pass  to  the  limit  e  =  0.  The  limit  of  Xi,  for  e  =  0,  is 
easily  proved  to  be  zero  by  using  polar  coordinates.  Hence  it 
follows  from  (12)  that 

lim  X2  =  X, 


e=0 


and  consequently,  from  the  last  inequalities, 


The  corresponding  relations  for  derivatives  with  respect  to  y 
and  z  are  proved  similarly,  and  therefore  equations  (11)  hold 
whether  or  not  P  is  a  part  of  M. 


75]  POTENTIAL  AND   ATTRACTION   OF   CIRCULAR  DISC.  113 

74.  Level  Surfaces.  The  equation  V  =  c,  where  c  takes  con- 
stant values,  defines  what  are  called  level  surfaces  or  equipotential 
surfaces. 

Any  displacement  bx,  dy,  5z,  of  the  particle  from  the  point 
(XQ,  i/o,  2o)  in  a  level  surface  must  fulfill  the  equation 


which  is  the  condition  that  the  points  (XQ,  T/O,  2o)  and  (XQ  -f  dx, 
2/o  +  8y,  ZQ  +  8z)  shall  both  be  in  the  same  level  surface.  This 
equation  becomes  as  a  consequence  of  (11) 

(14)  Xdx  +  Ydy  +  Zdz  =  0. 

The  direction  cosines  of  the  resultant  attraction  to  which  the 
particle  is  subject  are  proportional  to  X,  Y,  Z,  and  the  direction 
cosines  of  the  line  of  the  displacement  are  proportional  to  8x} 
8y,  8z.  Since  the  sum  of  the  products  of  these  direction  cosines 
in  corresponding  pairs  is  zero,  it  follows  that  the  resultant  attrac- 
tion is  perpendicular  to  the  level  surfaces.  Consequently,  if  the 
particle  starts  from  rest  it  will  begin  to  move  perpendicularly  to 
the  level  surface  through-  its  initial  position;  but  after  it  has 
acquired  an  appreciable  velocity  it  will  not  in  general  move 
perpendicularly  to  the  level  surfaces  because  the  motion  depends 
not  only  upon  the  forces,  which  have  been  shown  to  be  orthogonal 
to  the  level  surfaces,  but  also  upon  the  velocity. 

75.  The  Potential  and  Attraction  of  a  Thin  Homogeneous 
Circular  Disc  upon  a  Particle  in  its  Axis.  Take  the  origin  at  the 
center  of  the  disc  whose  radius  is  R.  Let  the  coordinates  of  P  be 
x,  0,  0.  Then 

Cdm  CR    C2«    rdrdd 

V  =    I   =  a   I  — -. _  • 

J     P  Jo     Jo      Ate2  +  r2 

Upon  integrating,  it  is  found  that 

[  V  =  27r<r[  Vz2  +  R2~  Vz2], 

(15)  ^   ^       ^dV       1  ,.    T        x x_-\ 


If  x  is  kept  constant  and  R  is  made  to  approach  infinity  as  a  limit, 
the  attraction  becomes 

(16)  X  =  =t=  27rA;2(7, 

9 


POTENTIAL   AND   ATTRACTION   OF 


[76 


according  as  the  particle  is  on  the  positive  or  negative  side  of  the  yz- 
plane.  The  right  member  of  this  equation  does  not  depend  upon  x; 
therefore  a  thin  circular  disc  of  infinite  extent  attracts  a  particle 
above  it  with  a  force  which  is  independent  of  its  altitude.  Any 


Fig.  19. 

number  of  superposed  discs  would  act  jointly  in  the  same  manner. 
Hence,  if  the  earth  were  a  plane  of  infinite  extent,  as  the  ancients 
commonly  supposed,  bodies  would  gravitate  toward  it  with 
constant  forces  at  all  altitudes,  and  the  laws  of  falling  bodies 
derived  under  the  hypothesis  of  constant  acceleration  would  be 
rigorously  true. 

76.  The   Potential  and  Attraction  of  a  Thin  Homogeneous 
Spherical  Shell  upon  an  Interior  or  an  Exterior  Particle.     Let 


(ff.O.O) 


Fig.  20. 

0  represent  the  angle  between  OP  and  the  radius,  and  6*  the  angle 
between  the  fundamental  plane  and  the  plane  OAP.     Then 


(17)  V  =    f  —  =  a-  C  C 


*  It  must  be  noticed  that  the  0  and  6  here  are  not  the  ordinary  polar  angles 
used  elsewhere. 


76]  THIN   HOMOGENEOUS   SPHERICAL   SHELL.  115 

One  of  the  three  variables  0,  6,  p  must  be  expressed  in  terms  of 
the  remaining  two.     From  the  figure  it  is  seen  that 

p2  =  x2  +  R2  -  2xR  cos  0; 
whence 

(18)  pdp  =  xR  sin 

Then  (17)  becomes,  if  P  is  exterior, 


rtir 

-,  1   dpde' 


and  if  P  is  interior, 


The  integrals  of  these  equations  are  respectively 

M 


M 
R' 

The  z-components  of  attraction  are  respectively 

i^U^...-** 

(22) 


which  agree  with  the  results  obtained  in  Arts.  66  and  70. 

The  attraction  of  a  solid  homogeneous  sphere  also  can  be  found 
at  once.  Considering  the  shell  as  an  element  of  the  sphere,  the 
M  of  (22)  is  given  by  the  equation 

M  = 


Let  X  represent  the  attraction  of  the  whole  sphere  M\  then 


. 

3      z2  x2 

Consider  the  mutual  attraction  of  two  spheres.  In  accordance 
with  the  results  which  have  just  been  obtained,  each  one  attracts 
every  particle  of  the  other  as  it  would  if  its  mass  were  all  at  its 
center.  Hence  the  two  spheres  attract  each  other  as  they  would 
if  their  masses  were  all  at  their  respective  centers. 

77.  Second  Method  of  Computing  the  Attraction  of  a  Homo- 
geneous Sphere.  A  very  simple  method  will  now  be  given  of 
finding  the  attraction  of  a  solid  homogeneous  sphere  upon  an 


116 


SECOND  METHOD   OF   COMPUTING 


[77 


exterior  particle  when  it  is  known  for  interior  particles.  It  is  a 
trivial  matter  in  this  case  and  is  introduced  only  because  the 
corresponding  device  in  the  much  more  difficult  case  of  the  attrac- 
tions of  ellipsoids  is  of  the  greatest  value,  and  constitutes  Ivory's 
celebrated  method. 

Let  it  be  required  to  find  the  attraction  of  the  sphere  S  upon 
the  exterior  particle  P',  supposing  it  is  known  how  to  find  the 
attraction  upon  interior  particles.  Construct  the  concentric 


Fig.  21. 

sphere  S'  through  P'  and  suppose  it  has  the  same  density  as  S. 
A  one-to-one  correspondence  between  the  points  on  the  surfaces 
of  the  two  spheres  is  established  by  the  relations 


/nn\ 

(23) 


x  =     7 


z  =  -7z. 


The  corresponding  points  are  in  lines  passing  through  the  common 
center  of  the  spheres,  and  P  corresponds  to  P'.  Let  X  and  X' 
represent  the  attractions  of  S'  and  S  upon  P  and  P'  respectively. 
They  are  given  by  the  equations 


(24) 


X  =  -  k2  C   ^—^- dm' = -k*<r  C  C  C^—^-dx'dy'dz', 
Joso     p  J  J  J     P 


Jw       P3 
But  it  follows  from  the  definition  of  p  and  p'  that 


(25)- 


77]  ATTRACTION  OF  HOMOGENEOUS  SPHERE.  117 

where  p2  and  pi  are  the  extreme  values  of  p  obtained  by  integrating 
with  respect  to  x.  That  is,  the  first  integration  gives  the  attrac- 
tion of  an  elementary  column  extending  through  the  sphere  parallel 
to  the  X-axis,  and  pi  and  p2  are  the  distances  from  the  attracted 
particle  P'  to  the  ends  of  this  column.  In  completing  the  inte- 
gration the  sum  of  all  of  these  elementary  columns  is  taken.  The 
corresponding  statements  with  respect  to  the  first  equation  of  (25) 
are  true. 

Suppose  the  integrals  (25)  are  computed  in  such  a  manner  that 
corresponding  columns  of  the  two  spheres  are  always  taken  at 
the  same  time.  Consider  any  two  pairs  of  corresponding  elements, 
as  those  at  A  and  A' '.  For  these  p  =  p',  and  this  relation  holds 
throughout  the  integration  as  arranged  above.  Hence  it  follows 
from  equations  (24)  and  (25)  that 

rr/i     i  \  c c i i      i  \ 

Xf  =  —  k2v  I          )  dydz  =  —  k2a  I       I  —. -.  }  dydz. 

J  J    \P2      PI/  J  J    \P2       PI  / 

But,  from  (23), 

R  R 

therefore 

R*JJ    \Pz       Pi'/  R'2 

Let  M  represent  the  mass  of  the  sphere  S,  and  M'  that  of  £'. 
The  attraction  of  Sf  upon  the  interior  particle  P  is  given  by 

X  =  - 


therefore   it   follows   from   the   relation   R'*Xf  =  R2X  that   the 
attraction  of  S  upon  the  exterior  particle  P'  is 


agreeing  with  results  previously  obtained  (Arts.  69,  70). 


118  PROBLEMS. 


XI.     PROBLEMS. 

1.  Prove  by  the  limiting  process  that  the  potential  and  components  of 
attraction  have  finite,  determinate  values,  and  that  equations  (11)  hold  when 
the  particle  is  on  the  surface  of  the  attracting  mass. 

2.  Find  the  expression  for  the  potential  function  for  a  particle  exterior  to 
the  attracting  body  when  the  force  varies  inversely  as  the  nth  power  of  the 

distance. 

,r  1        C     dm 

Y  - 


3.  Find  by  the  limiting  process  for  what  values  of  n  the  potential  in  the 
last  problem  is  finite  and  determinate  when  the  particle  is  a  part  of  the  at- 
tracting mass. 

4.  Show  that  the  level  surfaces  for  a  straight  homogeneous  rod  are  prolate 
spheroids  whose  foci  are  the  extremities  of  the  rod. 

5.  Find  the  components  of  attraction  of  a  uniform  hemisphere,  whose 
radius  is  R,  upon  a  particle  on  its  edge:  (a)  in  the  direction  of  the  center  of 
its  base;  (6)  perpendicular  to  this  direction  in  the  plane  of  the  base;  (c)  per- 
pendicular to  these  two  directions. 

Am.     (a)  X  =  liraWR',     (&)  7  =  0;     (c)  Z  =  &k*R. 

6.  Find  the  deviation  of  the  plumb-line  due  to  a  hemispherical  hill  of 
radius  r  and  density  a\.     Let  R  represent  the  radius  of  the  earth,  assumed 
to  be  spherical,  and  <r2  its  mean  density. 

Ans.     If  X  is  the  angle  of  deviation, 


-       .  D  .  -  -    , 

f  7TO-2/C  —  f  o\r       wa-iR  —  air 
or 

tan  X  =  -  —  -^  approximately. 

7.  Prove  that  if  the  attraction  varies  directly  as  the  distance,  a  body  of 
any  shape  attracts  a  particle  as  though  its  whole  mass  were  concentrated  at 
its  center  of  mass. 


78]     POTENTIAL  AND  ATTRACTION  OF  OBLATE  SPHEROID.    119 


78.  The  Potential  and  Attraction  of  a  Solid  Homogeneous 
Oblate  Spheroid  upon  a  Distant  Unit  Particle.  The  planets  are 
very  nearly  oblate  spheroids,  and  they  are  so  nearly  homogeneous 
that  the  results  obtained  in  this  article  will  represent  the  actual  facts 
with  sufficient  approximation  for  most  astronomical  applications. 

Suppose  the  attracted  particle  is  remote  compared  to  the 
dimensions  of  the  attracting  spheroid.  Take  the  origin  of  co- 


P  (x,  y,  z) 


Fig.  22. 

ordinates  at  the  center  of  the  spheroid  with  the  2-axis  coinciding 
with  the  axis  of  revolution.  Let  R  represent  the  distance  from 
0  to  P,  and  r  the  distance  from  0  to  the  element  of  mass.  Then 

dm 


(26) 


=   CdJH 

J(S)    p 


p  = 
R  = 


-  £)2  +  (y  -  r?)2  +(z- 


+  y2  4- 


I  r  =  V?  +  r?2  +  r2- 
It  follows  from  these  equations  that 

1  =  1 = 

P       V#2  +  r2  — 


1 


1  + 


r2  -  2(sg  +  yjj  + 
R2 


-5 ,  15 ,  and  j^  be  taken  as  small  quantities  of  the  first 
K     K  K 

order;  then,  on  expanding  the  expression  for  p"1  by  the  binomial 
theorem,  it  is  found  that,  up  to  small  quantities  of  the  third  order, 

+  vn  +  zf       r2 


P  =  £< 


120          POTENTIAL   AND   ATTRACTION    OF   OBLATE   SPHEROID.  [78 

Therefore 


(27) 


+5/1*  *•+;-•'• 

Let  M  represent  the  mass  of  the  spheroid;  then 
J  dm  =  M, 

and,  since  the  origin  is  at  the  center  of  gravity, 

/r  r 

£dm  =  0,         I  rjdm  =  0,         I  f  dm  =  0. 
J  J 

Let  0-  represent  the  density;  then 

dm  =  or2  cos  (f)d<f>d0dr, 
%  =  r  cos  0  cos  0, 
77  =  r  cos  0  sin  0, 

and  (27)  becomes 


cos3  0  sin  (9  cos  Bd^dBdr 
f  T  Tr4  sin  0  cos2  0  smdd<t>dddr 

sin  *  cos2  ^  cos 


where  the  limits  of  integration  are:  for  r,  0  and  r;  for  </>;  —  -  and 

~  ;  and  for  0,  0  and  2?r.     Since  r  and  0  are  independent  of  0,  the 
2i 

integration  can  be  performed  with  respect  to  0  first,  giving 


78]  POTENTIAL  AND   ATTRACTION   OF   OBLATE    SPHEROID.  121 


(28) 


+ 


TT 

r4  si 
J_JL  Jo 


sin2  4>  cos  4>d(f>dr  -f 


the  last  three  integrals  being  zero. 

The  next  integration  must  be  made  with  respect  to  r,  as  this 
variable  depends  upon  <f>.  Let  the  major  and  minor  semi-axes 
of  a  meridian  section  of  the  spheroid  be  a  and  b  respectively,  and 
let  e  be  the  eccentricity.  Then 


1  -  e2  cos2  </> ' 

Upon  integrating  (28)  with  respect  to  r  and  expanding  in  powers 
of  e,  it  is  found  that,  up  to  terms  of  the  second  order  inclusive, 


=  M 
=  R 


IT 

I  *  (1  +  fe2  cos2 <£  +  •  •  •)  cos  </>d<£ 


COS3  0C?0 


cos2  0  +  •  •  •)  sin2  0  cos<j>d<t> 


On  integrating  with  respect  to  <f>  and  arranging  in  powers  of 
the  expression  for  V  becomes 


But 


M  = 


122  POTENTIAL  AND   ATTRACTION   OF  [79 

therefore 

o»     '- 


The  components  of  attraction  are  found  from  equations  (11)  and 
(29)  to  be 


(30) 

Z=  -  ^ft  ["  1  +  ^V3(X*  +  |24}  ~  2*2  e2  +  •  •  •  1  . 

If  the  spheroid  should  become  a  sphere  of  the  same  mass,  the 
expressions  for  the  components  of  attraction  would  reduce  to  the 
first  terms  of  the  right  members  of  equations  (30) .  If  the  attracted 
particle  is  in  the  plane  of  the  equator  of  the  attracting  spheroid, 
2  =  0;  and  if  it  is  in  the  polar  line,  x  =  y  =  0.  Hence  it  follows 
from  (30)  that  the  attraction  of  an  oblate  spheroid  upon  a  particle 
at  a  given  distance  from  the  center  in  the  plane  of  its  equator  is  greater 
than  that  of  a  sphere  of  equal  mass;  and  in  the  polar  line,  less  than  that 
of  a  sphere  of  equal  mass.  As  the  particle  recedes  from  the  at- 
tracting body  the  attraction  approaches  that  of  a  sphere  of  equal 
mass.  Therefore,  as  the  particle  recedes  in  the  plane  of  the  equator 
the  attraction  decreases  more  rapidly  than  the  square  of  the  distance 
increases;  and  as  it  approaches,  the  attraction  increases  more  rapidly 
than  the  square  of  the  distance  decreases.  The  opposite  results  are 
true  when  the  particle  is  in  the  polar  line. 

79.  The  Potential  and  Attraction  of  a  Solid  Homogeneous 
Ellipsoid  upon  a  Unit  Particle  in  its  Interior.  Let  the  equation 
of  the  surface  of  the  ellipsoid  be 


and  let  the  attracted  particle  be  situated  at  the  interior  point 
(x,  y,  z).  Take  this  point  for  the  origin  of  the  polar  coordinates 
p,  6,  and  <f>.  On  taking  the  fundamental  planes  of  this  system 
parallel  to  those  of  the  first  system,  these  variables  are  related  to 
the  rectangular  coordinates  by  the  equations 


79] 


A   SOLID  HOMOGENEOUS  ELLIPSOID. 


123 


(£  =  x  +  p  cos  <f>  cos  6, 
TI  =  y  +  p  cos  0  sin  0, 
f  =  z  +  p  sin  0. 
The  potential  of  the  ellipsoid  upon  the  unit  particle  P  is 

C   dm  fT  f2jr  p1 

7  =          —  =  o-  I  I     p  cos  0  d<j>  d6  dp. 

J(M)   P  J~Jo    Jo 

Since  the  value  of  p  depends  upon  the  polar  angles  the  integration 
must  be  made  first  with  respect  to  this  variable.  The  integration 
gives 


(33) 


=  I  f  *  J2ff 


V  = 


Pl2  cos  0  d<j>  de. 


To  express  pi  in  terms  of  the  polar  angles  substitute  (32)  in  (31); 
whence  it  is  found  that 


(34) 
where 


(35) 


APl2  +  25Pl  +  C  =  0, 
.   _  cos2  0  cos2  0      cos2  0  sin2  0      sin2  0 

-^  O  7~0  I  o 


R 


a;  cos  0  cos  ^      y  cos  0  sin  0      z  sin  0 
~~  ~~  ~~ 


From  (34)  it  is  found  that 


Pl= 


-B 


The  only  pi  having  a  meaning  in  this  problem  is  positive;  A  is 
essentially  positive,  and  C  is  negative  because  (a;,  y,  z)  is  within 
the  ellipsoidal  surface.  Therefore  the  positive  sign  must  be 
taken  before  the  radical.  On  substituting  this  value  of  pi  in  (33), 
it  is  found  that 


*    n    f 
=  2Jn) 


foa\ 

(36) 


Consider  the  integral 


-  AC) 


124 


POTENTIAL  AND   ATTRACTION   OF 


[79 


It  follows  from  the  expression  for  B  that  the  differential  elements 
corresponding  to  6  =  00,  <t>  =  4>o  and  to  6  =  TT  +  00,  </>  =  —  <£o  are 
equal  in  numerical  value  but  opposite  in  sign.  Since  all  the 
elements  entering  in  the  integral  can  be  paired  in  this  way,  it 
follows  that  Vi  =  0,  after  which  (36)  becomes 


(37)  H 


iiun 


COS2  0  COS2 


?~c) 


X  (  -nr  - 


cos2  0  sin2  0 


cos  <f>  d<j>  dd 


s  (f>  sin  0  cos  B      yz  sin  0  cos  0  sin  B 

zx  sin  0  cos  0  cos  0  1  cos  <ft  d</>  c?0 
c2a2  ^^         ' 


By  comparing  the  elements  properly  paired,  it  is  seen  that  the 
second  integral  is  zero. 
Let 


(38)     TF=.£  f2    f2" cos<t>d<j>dd 

2  J_5_  Jo      cos2  0  cos2  0      cos2  4>  sh 

2  -  |  — 


0  sn2  0      sin2  <j> ' 
~~ 


then  (37)  can  be  written  in  the  form 
(39)  V  =  - 


a    da        b    db 


dc 


For  a  given  ellipsoid  W  is  a  constant,  and  the  equation  of  the 
level  surfaces  has  the  form 

Ciz2  +  C2y2  +  C3z2  =  constant, 

which  is  the  equation  of  concentric  similar  ellipsoids,  whose  axes 
are  proportional  to  Ci~*,  C-r*,  and  Cs~*. 

In  order  to  reduce  W  to  an  integrable  form,  let 


(40) 
then  (38)  becomes 


,,  _  cos20 
~~ 


cos2(/> 


79]  A   SOLID  HOMOGENEOUS  ELLIPSOID.  125 

w  _  <L  f^    f2ir         cos<j>d<f>de 

~  2  J_2L  Jo        M  COs2  °  +  N  Sin'  0 


-J1/ 

«/o      «/o 


cos  4>d(j>d0 


M  cos2  0  +  N  sin2  0  ' 


M  and  N  are  independent  of  6]  hence,  on  integrating  with  respect 
to  this  variable,  it  is  found  that* 


(41) 


f 


cos  0  d<f> 


V  (a2  sin2  0 + c2  cos2  0)  (b2  sin2  0 + c2  cos2  0) 

To  return  to  the  symmetry  in  a,  6,  and  c  which  existed  in  (38), 
Jacobi  introduced  the  transformation 


Vc2  +  s  ' 
whence 


C-= 

Jo     V(a2 


TF  = 


On  forming  the  derivatives  with  respect  to  a,  6,  and  c,  and  substi- 
tuting in  (39),  it  follows  that 


V  =  iraabc  C  (  1  -  -£  --  r 
Jo     \         a2  -f  s      62 


c2  + 

<42> 


^/(a2   _{_   g)(£2  _j_   S)(C2   ^_   s) 

The  components  of  attraction  are 

2ir<rabcxk2  ds 


=  /c2—  =         C 
daT'        Jo      a 


F  —  ^2  ___  _ 
~~"     A/        . 


(a2  +  s)  V(a2  - 

ds 


(43) 


Equation  (41)  is  homogeneous  of  the  second  degree  in  a,  b, 
*  Letting  tan  0  =  x,  the  integral  reduces  to  one  of  the  standard  forms. 


(62  +  s)  V(a2  +  s)  (62  +  s)  (c2  +  s)  ' 

z  =  k2  ?¥.  =          C 

dz  '        Jo      C2  +  s 


126  PROBLEMS. 

and  c;  and  therefore  —  ,  — ,  — ,  computed  from  (39),  are  ho- 
mogeneous of  degree. zero  in  the  same  quantities.  It  follows, 
therefore,  that  if  a,  6,  and  c  are  increased  by  any  factor  v  the 
components  of  attraction  X,  F,  and  Z,  will  not  be  changed;  or, 
the  elliptic  homoeoid  contained  between  the  ellipsoidal  surfaces  whose 
axes  are  a,  b,  c  and  va,  vb,  and  vc  attracts  the  interior  particle  P 
equally  in  opposite  directions.  (Compare  Art.  67.) 

The  component  of  attraction,  X,  is  independent  of  y  and  z 
and  involves  x  to  the  first  degree;  therefore  the  x-component  of 
attraction  is  proportional  to  the  x-coordinate  of  the  particle  and  is 
constant  everywhere  within  the  ellipsoid  in  the  plane  £  =  x.  Similar 
results  are  true  for  the  two  other  coordinates. 

Suppose  the  notation  has  been  chosen  so  that  a  >  b  >  c. 
Then  (41)  can  be  put  in  the  normal  form  for  an  elliptic  integral 
of  the  first  kind  by  the  substitution 

c  u 

sm  <p  = 


2       a2 -& 
K    "STT-? 

which  gives 

(44)        w  =  27r(ra&c   r~°~          au 


W  -  c2 

This  integral  can  be  readily  computed,  when  K2  is  small,  by  ex- 
panding the  integrand  as  a  power  series  in  /c2  and  integrating 
term  by  term. 

XII.    PROBLEMS. 

1.  Discuss  the  level  surfaces  given  by  equation  (29). 

2.  Set  up  the  expressions  for  the  components  of  attraction  instead  of  that 
for  the  potential  as  in  Art.  79.     Determine  what  parts  of  the  integrals  vanish, 
integrate  with  respect  to  0,  and  show  that  the  results  are 

X  =  -  lirvbcxk2 


Q 
Z  =  - 


>l  (ft2  sin2  0  +  a2  cos2  0)  (c2  sin2  0  -f  a2  cos2  0) ' 

sin2  0  cos  0  d<f> 
V  (c2  sin2  0  +  62  cos2  0)  (a2~sin2  0  +  62"co&2  0) ' 


F  =  -  ivacayk*  C  -  -=  sin2  0  cos 

JQ 


C 

J0 


sin2  *  cos 


0      V  (a2  sin2  <f>  +  c2  cos2  0)  (62  sin2  tf>  +  c2  cos2  <£) 


80] 


IVORY'S  METHOD. 


127 


Hint.     Derive  the  results  for  Z,  and  since  it  is  immaterial  in  what  order 
the  axes  are  chosen,  derive  the  others  by  a  permutation  of  the  letters  a,  b,  c. 
3.  Transform  the  equations  of  problem  2  by 

b 


sin  <f> 


Va2  + 


Sin 


sin  0  =  — = 


respectively,  and  show  that  equations  (43)  result. 

4.  Show  that  the  potential  of  an  ellipsoid  upon  a  particle  at  its  center  is 

s 


Vo  =  icaabc 


=  W. 


5.  From  the  value  of  Vo  and  equations  (43)  derive  the  value  of  the  po- 
tential (42). 

6.  Transform  the  equations  of  problem  2  so  that  they  take  the  form 

u?du 


f 


7.  Integrate  equations  (28)  without  expanding  the  expression  for  r2  as  a 
power  series  in  e2. 

80.  The  Attraction  of  a  Solid  Homogeneous  Ellipsoid  upon  an 
Exterior  Particle.  Ivory's  Method.  The  integrals  become  so 
complicated  in  the  case  of  an  exterior  particle  that  the  components 
of  attraction  have  not  been  found  by  direct  integration  except  in 
series.  They  are  computed  indirectly  by  expressing  them  in 


Fig.  23. 

terms  of  the  components  of  attraction  of  a  related  ellipsoid  upon 

particles  in  its  interior.     This  artifice  constitutes  Ivory's  method.* 

Let  it  be  required  to  find  the  attraction  of  the  ellipsoid  E  upon 

the  exterior  particle  P'  at  the  point  (x',  y',  z').     Let  the  semi- 

*  Philosophical  Transactions,  1809. 


128 


ATTRACTION   OF  A   SOLID   ELLIPSOID. 


[80 


axes  of  E  be  a,  b,  and  c.  Construct  through  P'  an  ellipsoid  Ef, 
confocal  with  E,  with  the  semi-axes  a',  b',  c',  and  suppose  it  has 
the  same  density  as  E.  The  axes  of  the  two  ellipsoids  are  related 
by  the  equations 


(45) 


a   = 


V  = 


where  K  is  defined  by  the  equation 
(46) 


a2  + 


b2  +  K  '  c2  + 


-1  =  0. 


The  only  value  of  K  admissible  in  this  problem  is  real  and  positive. 
Equation  (46)  is  a  cubic  in  K  and  has  one  positive  and  two  negative 
roots;  for,  the  left  member  considered  as  a  function  of  K  is  negative 


/(/C)  axis 


K-axts 


Fig.  24. 


when  K  =  +  °° ;  positive,  when  K  =  0  (because  (x'}  y',  z')  is 
exterior  to  the  ellipsoid  E) ;  positive,  when  K  =  —  c2  +  e  (where  t 
is  a  very  small  positive  quantity);  negative,  when  K  =  —  c2  —  e; 
positive,  when  K  =  —  62  +  e;  negative,  when  K  =  —  62  —  e;  posi- 
tive, when  K  =  —  a2  +  e;  negative,  when  K  =  —  a2  —  e;  and  nega- 
tive when  K  =  —  oo .  The  graph  of  the  function  is  given  in  Fig.  24. 
When  the  positive  root  is  taken,  a',  b',  and  c'  are  determined 
uniquely. 

A  one-to-one  correspondence  between  the  points  upon  the  two 


80J 


IVORYS  METHOD. 


129 


ellipsoids  will  now  be  established  by  the  equations  (compare 
Art.  77) 

(47)         r-j'fc     *v-|*     r-|r. 

Let  P  be  the  point  corresponding  to  Pf.  It  will  be  shown  that 
the  attraction  of  E  upon  Pr  is  related  in  a  very  simple  manner  to 
that  of  E'  upon  P. 

Let  X,  F,  and  Z  represent  the  components  of  attraction  of  E' 
upon  the  interior  particle  P  at  the  point  (x,  y,  z).  They  can  be 
computed  by  the  methods  of  Art.  79,  and  will  be  supposed  known. 
Let  X',  y,  and  Z'  be  the  components  of  attraction  of  E  upon  P', 
which  are  required.  The  expressions  for  the  re-components  are 


X  =  - 


On  performing  the  integration  with  respect  to  £,  it  is  found  that 


(49) 


where  P2  and  pi  are  the  distances  from  P'  to  the  ends  of  the  ele- 
mentary column  obtained  by  integrating  with  respect  to  £.  The 
solution  is  completed  by  integrating  over  the  whole  surface  of  E. 
The  first  equation  of  (49)  is  interpreted  similarly. 

Now  X'  will  be  related  to  X  in  a  simple  manner  by  the  aid  of 
the  following  lemma: 

//  P  and  A  are  any  two  points  on  the  surface  of  E,  and  if  P'  and 
A'  are  the  respective  corresponding  points  on  the  surface  of  E',  then 
the  distances  PA'  and  P'A  are  equal. 

Let  ~PAf  =  p'  and  ~AP'  =  p.  Then  p  =  p'.  For,  let  the 
coordinates  of  P  and  A  be  respectively  £1,771,  Ti  and  £2,  r?2,  £2;  and 
of  P'  and  A',  fc',  Tj/,  f  /  and  fr',  172',  ft'.  Then 


+  (i?!  -  172')*  +  (ri  - 


P2  =  tta  - 


~  f  i') 


10 


130  ATTRACTION   OF   SOLID   ELLIPSOID.  [80 

On  making  use  of  equations  (45)  and  (47),  it  is  found  that 


Since  P  and  A  are  on  the  surface  of  the  ellipsoid  whose  semi-axes 
are  a,  6,  and  c,  each  parenthesis  equals  unity.  Therefore  p'2  —  p2  =  0, 
or  p  =  p'. 

Suppose  the  integrals  (49)  are  computed  so  that  the  elements  at 
corresponding  points  of  the  two  surfaces  are  always  taken  simul- 
taneously. Then  pi  =  p/  and  p2  =  p2'  throughout  the  integration. 

b  c 

Moreover,  it  follows  from  (47)  that  dy  =  r?  dy'  and  d£  =  -,  d$'. 

Therefore 


(60) 

and  similarly 

v,  _  ca  v 

c'n'      ' 

(51) 

Z'  =  —Z 

The  letters  a,  b,  c,  and  s  of  equations  (43)  should  be  given  accents 
to  agree  with  the  notations  of  this  article;  and,  since  P  and  P' 

are   corresponding  points,   x  =  —,  x',  y  =  r-f  y',  z  =  -,  z'.    After 

CL  \J  C 

making  these  changes  in  equations  (43)  and  substituting  them 
in  (50)  and  (51),  it  is  found  that 


X'  =  - 


(a* 
Y'  =  - 


c/2 


Z'  =  -  2<jraabck2z 


Jo     (r'2 


(c/2  +  8')  V(a'2  +  «')  (6/2  +  «0  (c/2 
It  follows  from  equations  (45)  that 

a/2  =  a2  +  K,        b'2  =  62  +  K,        c'2  =  c2  +  K; 


80]  IVORY'S  METHOD.  131 

hence,  on  letting  s  =  sf  +  K,  it  follows  that 
X'  =- 


(52) 


(a2  +  s)  V(a2  +  s)  (62  +  s)  (c2  +  s) 


Y'  =  -  2jr<rabck2y 


•f 

f 


(62  +  s)  V(a2  +  s)  (62  +  s)  (c2  + 


It  follows  from  equations  (40)  and  (41)  that  the  components 
of  attraction  for  interior  particles  are  homogeneous  of  degree  zero 
in  a,  b,  and  c,  and  that  they  are  proportional  to  the  respective 
coordinates  of  the  attracted  particle.  Let  X}  as  above,  represent 
the  attraction  of  the  ellipsoid  E' ,  whose  semi-axes  are  a',  b',  cf, 
upon  the  interior  particle  at  (x,  y,  z) ;  let  X"  represent  the  attrac- 
tion of  E'  upon  an  interior  particle  at  (x",  y",  z"),  which  will  be 
supposed  to  be  related  to  (x,  y,  z)  by  equations  of  the  same  form 
as  (47).  Then  it  follows  that 

~Y''~~~^'       ~T~'~~~y'       ~Z==7* 

Let  the  point  (x",  y",  z"},  always  corresponding  to  (x}  y,  z), 
approach  the  surface  of  E'  as  a  limit.  Then  at  the  limit 

T''=~a^'        T:=6"'        ~Z~==c* 
On  combining  these  equations  with  (50)  and  (51),  it  is  found  that 

^  =  II  =  ?1  =  °L^-  =  ML 
X'   :=  Y'  ~  Z'   "    abc    ''=  M' 

That  is,  the  attraction  of  a  solid  ellipsoid  upon  an  exterior  particle 
is  to  the  attraction  of  a  confocal  ellipsoid  passing  through  the  particle, 
as  the  mass  of  the  first  ellipsoid  is  to  that  of  the  second  ellipsoid. 

Consider  another  ellipsoid  confocal  with  the  one  passing  through 
the  particle  and  interior  to  it;  by  the  same  reasoning  the  ratios 
of  the  components  of  attraction  of  these  two  ellipsoids  are  as 
their  masses.  Let  X'",  Y"',  Z'"  be  the  components  of  attraction 
of  the  new  ellipsoid  whose  semi-axes  are  a'",  b'",  c'".  Then 


X'"  ~  Y'"  ~  Z'"  ~  a"fb'"cf"  ~  M'" ' 
On  combining  this  proportion  with  (53),  it  is  found  that 


132 


THE  ATTRACTION   OF   SPHEROIDS. 


[81 


X' 

X"' 


Z' 

Z'" 


__ 
M'"' 


Therefore,  two  confocal  ellipsoids  attract  particles  which  are  exterior 
to  both  of  them  in  the  same  direction  and  with  forces  which  are  pro- 
portional to  their  masses.  This  theorem  was  found  by  Maclaurin 
and  Lagrange  for  ellipsoids  of  revolution,  and  was  extended  by 
Laplace  to  the  general  case  where  the  three  axes  are  unequal. 
It  is  established  most  easily,  however,  by  Ivory's  method  as  above, 
and  it  is  frequently  called  Ivory's  theorem. 

The  right  members  of  equations  (52)  can  be  transformed  to 
forms  which  are  more  convenient  for  computation  by  putting,  in 

the  first,     .  =  u',  in  the  second,  —  =  u; 


and  in  the 


third, 


=  u. 


The  results  of  the  substitutions  are 


(54) 


X'=-±Trabck2x' 


a 

'*!**+*. 


uzdu 


V[a2  -  (a2- 


v?du 


o  V[c2  -  (c2  -  a2)^2][c2  -  (c2-62K] 

When  the  attracted  particle  is  in  the  interior  of  the  ellipsoid  the 
forms  of  the  integrals  are  the  same  except  that  the  upper  limits  are 
unity. 

81.  The  Attraction  of  Spheroids.  The  components  of  attraction 
will  be  obtained  from  (54),  which  hold  for  exterior  particles. 
Suppose  the  attracting  body  is  an  oblate  spheroid  in  which  a  =  b>c 
and  let  e  represent  the  eccentricity  of  a  meridian  section.  Then 

c2  =  a2(l  -  e2), 
and  equations  (54)  become 


(55) 


The  integrals  of  these  equations  are 


82]  ATTRACTION  AT  THE  SURFACES  OF  SPHEROIDS. 


133 


[X1      Y' 

0         7.2     *                ^                                        /1                tt  ^ 

x'  '-  y'  ~ 

zr  _     4?r 

—  ZTTffK    ~   —  -£  |    2      ,  —  A  /  1   g      , 

•  |  •  sin     i     .  
W\      ce           vr—  ; 

? 

r    .;  I      .  —                 \i         o 

^LVc2H-K 
v  f  1  /                ce 

(56)    , 


(l_e2)(c2   +  K) 

The  components  of  attraction  for  interior  particles  are  obtained 
from  equations  (56)  by  putting  K  =  0. 

Now  suppose  the  attracting  body  is  a  prolate  spheroid  and 
that  a  =  b  <  c.  Then  a2  =  62  =  c2(l  -  e2),  and  equations  (64) 
become 


(57) 


J 


^  =  -  4T<rfc2(l  - 


The  integrals  of  these  equations  are 

^  =  7  = 


(58)  H 


c3    L 

(1  -  e») 

^  J_         -"—         £       ^ 

a,          r 

2      1 

a2e2 

A^T-.V1 

€^  H-      , 

a 
ae 

•+. 

log(v(r: 

*il 

,/    -2«  ^ 

-  e2)(a2  + 

^ 

i    i 

oV 

f(l- 
1+- 

e2)(a2  +  ic) 

C6          V 

«* 

1  - 

«             ' 

)• 


When  the  particle  is  interior  to  the  spheroid  the  equations  for 
the  components  of  attraction  are  the  same  except  that  K  =  0. 

82.  The  Attraction  at  the  Surfaces  of  Spheroids.  The  com- 
ponents of  attraction  for  an  interior  particle,  which  are  obtained 
in  the  case  of  an  oblate  spheroid  from  (56)  by  putting  K  =  0, 
are,  omitting  the  accents, 


134 


ATTRACTION  AT  THE  SURFACES  OF  SPHEROIDS. 


[82 


(59)  H 


-  =  —  =  -  27r<r/b2 


- 

x 

f 


-  e 


-  e2  +  sin-1  e], 


The  limits  of  these  expressions  as  the  attracted  particle  approaches 
the  surface  of  the  spheroid  are  the  components  of  attraction  for  a 
particle  at  the  surface.  As  the  attracted  particle  passes  outward 
through  the  surface,  K,  in  equations  (56),  starts  with  the  value 
zero  and  increases  continuously  in  such  a  manner  that  it  always 
fulfills  equation  (46).  Therefore  equations  (59),  having  no 
discontinuity  as  the  attracted  particle  reaches  the  surface,  hold 
when  x,  y,  z  fulfill  the  equation  of  the  ellipsoid. 

When  e  is  small,  as  in  the  case  of  the  planets,  equations  (59) 
are  convenient  when  expanded  as  power  series  in  e.  On  substi- 
tuting the  expansions 


sm"1  e  = 


in  equations  (59),  it  is  found  that 
X=  Y 

(60) 


The  mass  of  the  spheroid  is 

M  =  %iraa?c  = 


^ ..  v  *,  v  -..»—      if  A  c/  • 

The  radius  of  a  sphere  having  equal  mass  is  defined  by  the  equation 

M  =  ^TTffR3  =  -£7rcra3Vl  —  e2; 
whence 

R  =  a(l  -  e2)*. 

The  attraction  of  this  sphere  for  a  particle  upon  its  surface  is 
given  by  the  equation 

(61)  F  =  - 


82]  ATTRACTION   AT  THE   SURFACES  OF   SPHEROIDS.  135 

When  the  attracted  particle  is  at  the  equator  of  the  spheroid 
Vz2  +  y2  =  a;  hence  the  ratio  of  the  attraction  of  the  spheroid 
for  a  particle  at  its  equator  to  that  of  an  equal  sphere  for  a  particle 
upon  its  surface  is 


VZ2+  F2_(l  ~  &*  ..-)  <?_ 

—p-  (1-e2)1  30  "^ 

This  is  less  than  unity  when  e  is  small;  therefore  the  attraction  of 
the  spheroid  for  a  particle  on  its  surface  at  its  equator  is  less 
than  that  of  a  sphere  having  equal  mass  and  volume  for  a  particle 
on  its  surface. 

When  the  attracted  particle  is  at  the  pole  of  the  spheroid 
z  =  c  =  a  Vl  —  e2;  hence  in  this  case 


=  , 

(I  -  e2)*  r  15  ^ 

This  is  greater  than  unity  when  e  is  small;  therefore  the  attraction 
of  the  spheroid  for  a  particle  on  its  surface  at  its  pole  is  greater 
than  that  of  a  sphere  having  equal  mass  and  volume  for  a  particle 
on  its  surface. 

There  is  some  place  between  the  equator  and  pole  at  which  the 
attractions  are  just  equal.  The  latitude  of  this  place  will  now 
be  found.  The  coordinates  of  the  particle  must  fulfill  the  equa- 
tion of  the  spheroid;  therefore 

(62)  f(x,y,z)^^±^  +     -l=0. 


The  direction  cosines  of  the  normal  to  the  surface  at  the  point 
(a?,  y,  z)  are 

*L  £ 

dx  dy 


dz 


/#Y 

\dz) 


The  last  is  the  cosine  of  the  angle  between  the  normal  at  the 
point  (x,  y,  z)  and  the  z-axis,  and  is,  therefore,  the  sine  of  the 


136 


ATTRACTION   AT   THE    SURFACES   OF   SPHEROIDS. 


[82 


geographical  latitude,  which  will  be  represented  by  <£.     Hence, 
it  follows  from  (62)  that 


(63) 


dz 


sin 


From  (62)  and  (63)  it  is  found  that 
a2  cos2  <b 


I  -  e2  sin2 


z2  =  a2(l  -  e2)2sin24> 


(64)  < 


Let  G  represent  the  whole  attraction  of  the  spheroid;  then  it  is 
found  from  (60)  and  (64)  that 


G  =  - 


+  Y2  +  Z2 


-  cos 


The  ratio  of  this  expression  to  that  for  the  attraction  of  a  sphere 
of  equal  mass  and  volume,  given  by  (61),  is 


(65)          = 


=  1 


-  3  sin2 


30 


This  becomes  equal  to  unity  up  to  terms  of  the  fourth  order  in  e 
when  3  sin2  0  =  1,  from  which  it  is  found  that 

0  =  35°  15'  52". 
Let  r  represent  the  radius  of  the  spheroid;  then 

2  =     a2(l  -  e2) 
~  1  -  e2  cos2^' 

where  $  is  the  angle  between  the  radius  and  the  plane  of  the 
equator.  Since  this  angle  differs  from  <J>  only  by  terms  of  the 
second  and  higher  orders  in  e,  it  follows  that,  with  the  degree  of 
approximation  employed, 


PROBLEMS.  137 


When  0  =  35°  15'  52 


The  radius  of  a  sphere  of  equal  volume  has  been  found  to  be 
given  by  the  equation 


which  is  seen  to  be  equal  to  the  radius  of  the  spheroid  up  to  terms 
of  the  second  order  inclusive  in  the  eccentricity.  Therefore,  in 
the  case  of  an  oblate  spheroid  of  small  ecentricity,  the  intensity 
of  the  attraction  is  sensibly  the  same  for  a  particle  on  its  surface 
in  latitude  35°  15'  52"  as  that  of  a  sphere  having  equal  mass  and 
volume  for  a  particle  on  its  surface;  or,  because  of  the  equality 
of  R  and  r,  a  spheroid  of  small  eccentricity  attracts  a  particle  on 
its  surface  in  latitude  35°  15'  52"  with  sensibly  the  same  force  it 
would  exert  if  its  mass  were  all  at  its  center. 

Xm.    PROBLEMS. 

1.  Show  that  Ivory's  method  can  be  applied  when  the  attraction  varies 
as  any  power  of  the  distance. 

2.  Show  why  Ivory's  method  cannot  be  used  to  find  the  potential  of  a 
solid  ellipsoid  upon  an  exterior  particle  when  it  is  known  for  an  interior  particle. 

3.  Find  the  potential  of  a  thin  ellipsoidal  shell  contained  between  two 
similar  ellipsoids  upon  an  interior  particle.         Hint.     It  has  been  proved 
(Art.  79)  that  the  resultant  attraction  is  zero  at  all  interior  points;  therefore 
the  potential  is  constant  and  it  is  sufficient  to  find  it  for  the  center.     Let  the 
semi-axes  of  the  two  surfaces  be  a,  6,  c  and  (1  +  n}a,  (1  +  fj,)b,  (1  +  ju)c;  then 
the  distance  between  the  two  surfaces  measured  along  the  radius  from  the 
center  will  be  pp.     Therefore 


-lif 


r 


cos  (t>d<f>dd 


cos2  <ft  cos2  0       cos2  <f>  sin2  6      sin2 
~  +~ 


V(a2  +  s)(62  +  s)(c2  -f-s) 


138  HISTORICAL   SKETCH. 

4.  Show  that  in  the  case  of  two  thin  confocal  shells  similar  elements  of 
mass  at  points  which  correspond  according  to  the  definition  (47)  are  propor- 
tional to  the  products  of  the  three  axes  of  the  respective  ellipsoids.  Then 
show,  using  problem  3  and  Ivory's  method,  that  the  potential  of  an  ellipsoidal 
shell  upon  an  exterior  particle  is 

J«  ds' 

.     ,  ,  ,  ==- 

Va'2  +  s'&'   +c'        «' 


/•»  ds 

2ira-fjLabc  I          , 

J<      V(a2  • 


5.  Prove  that  the  level  surfaces  of  thin  homogeneous  ellipsoids  are  confocal 
ellipsoids.     What  are  the  lines  of  force  which  are  orthogonal  to  these  surfaces? 

6.  Discuss  the  form  of  level  surfaces  when  they  are  entirely  exterior  to 
homogeneous  solid  ellipsoids. 

HISTORICAL  SKETCH  AND  BIBLIOGRAPHY. 

The  attractions  of  bodies  were  first  investigated  by  Newton.  His  results 
are  given  in  the  Principia,  Book  i.,  Sees.  xn.  and  xin.,  and  are  derived  by 
synthetic  processes  similar  to  those  used  in  the  first  part  of  this  chapter. 
The  problem  of  the  attraction  of  ellipsoids  has  been  the  subject  of  many 
memoirs,  and  the  case  in  which  they  are  homogeneous  was  completely  solved 
early  in  the  nineteenth  century.  Among  the  important  papers  are  those 
by  Stirling,  1735,  Phil.  Trans.;  by  Euler,  1738,  Petersburg;  by  Lagrange, 
1773  and  1775,  Coll  Works,  vol.  in.,  p.  619;  by  Laplace,  1782,  Mec.  Cel., 
vol.  ii.;  by  Ivory,  1809-1828,  Phil.  Trans.;  by  Legendre,  1811,  Mem.  de 
VInst.  de  France,  vol.  XL;  by  Gauss,  Coll.  Works,  vol.  v.;  by  Rodriguez,  1816, 
Corres.  sur  I'Ecole  Poly.,  vol.  in.;  by  Poisson,  1829,  Conn,  des  Tem,ps;  by 
Green,  1835,  Math.  Papers,  vol.  vin.;  Chasles,  1837-1846,  Jour.  I'Ecole 
Poly,  and  Mem.  des  Savants  Strangers,  vol.  ix.;  MacCullagh,  1847,  Dublin 
Proc.,  vol.  in.;  Lejeune-Dirichlet,  Journal  de  Liouville,  vol.  iv.,  and  Crelle, 
vol.  xxxii. 

The  earlier  papers  were  devoted  for  the  most  part  to  the  attractions  of 
homogeneous  ellipsoids  of  revolution  upon  particles  in  particular  positions, 
as  on  the  axis.  Lagrange  gave  the  general  solution  for  the  attractions  of 
general  homogeneous  ellipsoids  upon  interior  particles.  This  was  extended 
by  Ivory  and  Maclaurin  (with  Laplace's  generalizations)  to  exterior  particles. 
Ivory's  theorem  has  been  extended  in  a  most  interesting  manner  by  Darboux 
in  Note  xvi.  to  the  second  volume  of  the  Mecanique  of  Despeyrous.  Chasles 
gave  a  synthetic  proof  of  the  theorems  regarding  the  attractions  of  homo- 
geneous ellipsoids  in  Memoir es  des  Savants  Strangers,  vol.  ix.,  and  Lejeune- 
Dirichlet  embraced  in  a  most  elegant  manner  in  one  discussion  the  case  of 
both  interior  and  exterior  points  by  using  a  discontinuous  factor  (Liouville's 
Journal,  vol.  iv.). 

Laplace  proved  that  the  potential  for  an  exterior  particle  fulfills  the  partial 
differential  equation 

Sr+*r.£r- 

ar2  +  d*  +  a?  "'  u' 


HISTORICAL  SKETCH. 


139 


and  determined  V  by  the  condition  that  it  must  be  a  function  satisfying  this 
equation.  This  is  a  process  of  great  generality,  and  is  relatively  simple 
except  in  the  trivial  cases.  This  has  been  made  the  starting-point  of  most 
of  the  investigations  of  the  latter  part  of  the  last  century,  especially  where 
the  attracting  bodies  are  not  homogeneous.  In  a  paper  on  Electricity  and 
Magnetism,  in  1828,  Green  introduced  the  term  potential  function  for  V,  and 
discussed  many  of  its  mathematical  properties.  Green's  memoir  remained 
nearly  unknown  until  about  1846,  and  in  the  meantime  many  of  his  theo- 
rems had  been  rediscovered  by  Chasles,  Gauss,  Sturm,  and  Thomson.  One 
of  Green's  theorems  has  found  an  extremely  useful  application,  when  the 
independent  variables  are  two  in  number,  in  the  Theory  of  Functions. 

Poisson  showed  that  the  potential  function  for  an  interior  particle  fulfills 
the  partial  differential  equation 


Among  the  books  treating  the  subject  of  attractions  and  potential  may  be 
mentioned  Thomson  and  Tait's  Natural  Philosophy,  part  u.,  Neumann's 
Potential,  Poincar6's  Potential,  Routh's  Analytical  Statics,  vol.  n.,  and  Tisser- 
and's  Mecanique  Celeste,  vol.  n.  The  last-mentioned  develops  most  fully 
the  astronomical  applications  and  should  be  used  in  further  reading. 

The  attractions  of  spheroids  and  ellipsoids  has  been  fundamental  in  the 
discussions  of  possible  figures  of  equilibrium  of  rotating  fluids.  The  reason 
is,  of  course,  that  the  conditions  for  equilibrium  involve  the  components  of  at- 
traction. Maclaurin  proved  in  1742  that  for  slow  rotation  an  oblate  spheroid, 
whose  eccentricity  is  a  function  of  the  rate  of  rotation  and  the  density  of  the 
fluid,  ia  a  figure  of  equilibrium.  There  are,  indeed,  two  such  figures;  for  slow 
rotation  one  is  nearly  spherical  and  the  other  is  very  much  flattened.  For 
faster  rotation  the  figures  are  more  nearly  of  the  same  shape;  for  a  certain 
greater  rate  of  rotation  they  are  identical;  and  for  still  faster  rotation  no 
spheroid  is  a  figure  of  equilibrium.  In  1834  Jacobi  proved  that  when  the  rate 
of  rotation  is  not  too  great  there  is  an  ellipsoid  of  three  unequal  axes  which  is  a 
figure  of  equilibrium,  which  for  a  certain  rate  of  rotation  coincides  with  the 
more  nearly  spherical  of  the  Maclaurin  spheroids.  For  this  work  Tisserand's 
Mecanique  Celeste,  vol.  n.,  should  be  consulted.  In  a  very  important  memoir 
(Acta  Mathematics,  vol.  vn.)  Poincare"  proved  that  there  are  infinitely  many 
other  figures  of  equilibrium  which,  for  certain  values  of  the  rate  of  rotation, 
coincide  with  the  corresponding  Jacobian  ellipsoid,  as  it,  for  a  certain  rate 
of  rotation,  coincides  with  the  Maclaurin  spheroid.  The  least  elongated  of 
these  figures  is  larger  at  one  end  than  it  is  at  the  other,  and  was  called  the 
apioid,  that  is,  the  pear-shaped  figure.  Later  computations  by  Sir  George 
Darwin  (Philosophical  Transactions,  vol.  198)  have  shown  it  is  so  elongated 
that  it  might  well  be  called  a  cucumber-shaped  figure. 


CHAPTER  V. 


THE  PROBLEM   OF   TWO  BODIES. 

83.  Equations  of  Motion.  It  will  be  assumed  in  this  chapter 
that  the  two  bodies  are  spheres  and  homogeneous  in  concentric 
layers.  Then,  in  accordance  with  the  results  obtained  in  Art.  69, 
they  will  attract  each  other  with  a  force  which  is  proportional  ,to 
the  product  of  their  masses  and  which  varies  inversely  as  the 
square  of  the  distance  between  their  centers. 

Let  mi  and  m2  represent  the  masses  of  the  two  bodies,  and 
mi  +  mz  =  M.  Choose  an  arbitrary  system  of  rectangular  axes 
in  space  and  let  the  coordinates  of  mi  and  w2  referred  to  it  be 
respectively  (£1,  171,  fi)  and  (£2,  ??2,  £2).  Let  the  distance  between 
mi  and  ra2  be  denoted  by  r;  then  it  follows  from  the  laws  of  motion 
and  the  law  of  gravitation  that  the  differential  equations  which  the 
coordinates  of  the  bodies  satisfy  are 


(1) 


dt2 

d2rji 
dt* 


dt* 


r6 


In  order  to  solve  these  six  simultaneous  equations  of  the  second 
order  twelve  integrals  must  be  found.  They  will  introduce  twelve 
arbitrary  constants  of  integration  which  can  be  determined  in  any 
particular  case  by  the  three  initial  coordinates  and  the  three  com- 
ponents of  the  initial  velocity  of  each  of  the  bodies. 

140 


84] 


THE   MOTION   OF   THE   CENTER   OF   MASS. 


141 


84.  The  Motion  of  the  Center  of  Mass.  On  adding  the  first 
and  fourth,  the  second  and  fifth,  and  the  third  and  sixth  equations 
of  the  system  (1),  it  is  found  that 


These  equations  are  immediately  integrable,  and  give 


(2) 


dt 


On  integrating  again  they  become 


dt  ~ 

=  ait 

=  Pit 


182, 
72. 


Thus,  six  of  the  twelve  integrals  are  found,  the  arbitrary  constants 
of  integration  be_ing  on,  o%  |8i,  j82,  71,  72- 

Let  I",  ^  i  and  p  be  the  coordinates  of  the  center  of  mass  of  the 
system;  then  it  follows  from  Art.  19  and  "equations  (3)  that 


(4) 


+ 


=  ait 

=  Pit 


/32, 


72. 


From  these  equations  it  follows  that  the  coordinates  increase 
directly  as  the  time,  and,  therefore,  that  the  center  of  mass  moves 
with  uniform  velocity.  Or,  taking  their  derivatives,  squaring, 
and  adding,  it  is  found  that 


'KfHfHfn 


7i2; 


whence 


142 


THE   EQUATIONS  FOR   RELATIVE   MOTION. 


[85 


=  _ 


M 


where  V  represents  the  speed  with  which  the  center  of  mass 
moves.     The  speed  is  therefore  constant. 
On  eliminating  t  from  (4),  it  is  found  that 


on  Pi  7i 

The  coordinates  of  the  center  of  mass  fulfill  these  relations  which 
are  the  symmetrical  equations  of  a  straight  line  in  space ;  therefore, 
the  center  of  mass  moves  in  a  straight  line  with  constant  speed. 

85.  The  Equations  for  Relative  Motion.  Take  a  new  system 
of  axes  parallel  to  the  old,  but  with  the  origin  at  the  center  of  mass 
of  the  two  bodies.  Let  the  coordinates  of  mi  and  m2  referred  to 
this  new  system  be  xi,  y\,  z\  and  x*,  yi,  22  respectively.  They 
are  related  to  the  old  coordinates  by  the  equations 


(5) 


=  rji  —  rj, 


£2  =  £2  -  £, 
2/2  =  rjz  —  rj, 


On  substituting  in  (1),  the  differential  equations  of  motion  in  the 
new  variables  are  found  to  be 


(6) 


—  22) 


dt2 


r3 

(22  - 


which  are  of  the  same  form  as  the  equations  for  absolute  motion. 

The  coordinates  of  the  center  of  mass  are  given  by  equations  (4) ; 

therefore  if  x\t  y\,   >>••>>,  z2  were  known,  and  if  the  constants 


85] 


THE   EQUATIONS   FOR   RELATIVE   MOTION. 


143 


«i,  "2,  j8i,  02,  7i>  and  72  were  known,  the  absolute  positions  in 
space  could  be  found.  But,  since  there  is  no  way  of  determining 
these  constants,  the  problem  of  relative  motion,  as  expressed 
in  (6),  is  all  that  can  be  solved. 

Since  the  new  origin  is  at  the  center  of  mass,  the  coordinates 
are  related  by  the  equations 


(7) 


=  0, 
=  0, 

=  0. 


Therefore,  when  the  coordinates  of  one  body  with  respect  to  the 
center  of  mass  of  the  two  are  known  the  coordinates  of  the  second 
body  are  given  by  equations  (7) . 

Equations  (7)  can  be  used  to  eliminate  x*,  2/2,  and  zz  from  the 
first  three  equations  of  (6),  and  x\,  y\,  and  z\  from  the  last  three. 
The  results  of  the  elimination  are 


(8) 


df 


dt2 

In  the  first  three  equations  the  r  which  appears  in  the  right 
member  must  be  expressed  in  terms  of  x\,  yi,  and  z\\  and  in  the 
second  three  it  must  be  expressed  in  terms  of  x2,  yz,  and  z2.  It 
follows  from  equations  (7)  that 


M 


M 


M 


r. 


The  equations  in  Xi,  y\,  z\  are  now  independent  of  those  in  z2, 2/2,  22, 
and  conversely.     But  what  is  really  desired  in  practice  is  the 


144 


THE    INTEGRALS    OF   AREAS. 


motion  of  one  body  with  respect  to  the  other.     Let  x,  y,  and  2 
represent  the  coordinates  of  ra2  with  respect  to  mi,  then 


xz  - 


z  =  zz  - 


Hence  if  the  first,  second,  and  third  equations  of  (8)  are  sub- 
tracted from  the  fourth,  fifth,  and  sixth  equations  respectively,  the 
results  are,  as  a  consequence  of  these  relations, 


(9) 


The  problem  is  now  of  the  sixth  order,  having  been  reduced 
from  the  twelfth  by  means  of  the  six  integrals  (2)  and  (3).  The 
six  new  constants  of  integration  which  will  be  introduced  in 
integrating  equations  (9)  will  be  determined  by  the  three  initial 
coordinates,  and  the  three  projections  of  the  initial  velocity  of  mi 
with  respect  to  m2. 

86.  The  Integrals  of  Areas.  Multiply  the  first  equation  of  (9) 
by  —  y,  and  the  second  by  +  x}  and  add;  the  result  is 


d?z        d?y 

-'5? 

dzx        d2z 


The  integrals  of  these  equations  are 

dy         dx 


(10) 


dz         dy 

di-zdi  =  a*' 

dx        dz 


_ 
Zdt       Xdt 


as. 


It  follows  from  Art.  16  that  a\,  a2,  a3  are  the  projections  of 
twice  the  areal  velocity  upon  the  xy,  yz,  and  zz-planes  respectively. 


86] 


THE    INTEGRALS    OF   AREAS. 


145 


Upon  multiplying  equations  (10)  by  2,  x,  and  y  respectively,  and 
adding,  it  is  found  that 

(11)  aiz  +  azx  +  a^y  =  0. 

This  is  the  equation  of  a  plane  passing  through  the  origin,  and  it 
follows  from  its  derivation  that  the  coordinates  of  mi  always 
fulfill  it ;  therefore,  the  motion  of  one  body  with  respect  to  the  other  is 
in  a  plane  which  passes  through  the  center  of  the  other. 

The  constants  01,  a2,  and  a3  determine  the  position  of  the 
plane  of  the  orbit  with  respect  to  the  axes  of  reference.  In  polar 
coordinates  equation  (11)  becomes 

(12)  ai  sin  <p  +  a%  cos  <p  cos  0  +  a3  cos  <p  sin  6  —  0. 

The  x7/-plane  and  the  plane  of  the  orbit  intersect  in  a  line  L 
(Fig.  25).  Suppose  OL  is  that  half  line  which  passes  through 


Fig.  25. 

the  point  at  which  the  body  mi  goes  from  the  negative  to  the 
positive  side  of  the  zi/-plane.  Let  &>  represent  the  angle  between 
the  positive  end  of  the  z-axis  and  the  line  OL  counted  in  the 
positive  direction  from  Ox.  This  angle  may  have  any  value  from 
0°  to  360°.  Let  i  represent  the  inclination  between  the  two 
planes  counted  in  the  direction  of  positive  rotation  around  OL. 
The  angle  i  may  have  any  value  from  0°  to  180°.  It  is  less  or 
greater  than  90°  according  as  ax  is  positive  or  negative.  Then, 
11 


146  PROBLEM   IN   THE    PLANE.  [87 

when  <p  =  0  the  value  of  6  is  ft  or  ft  +  IT.  When  6  =  ft  +  \ir 
the  value  of  <p  is  i  or  TT  —  i  according  as  i  is  less  than  or  greater 
than  90°.  In  these  cases  equation  (12)  becomes  respectively 

Ja2  cos  ft  +  a3  sin  ft  =  0, 
[ai  sin  i  =F  a2  cos  t  sin  &  =*=  as  cos  i  cos  ,0,  =  0, 

where  the  signs  of  the  second  equation  are  the  upper  if  i  is  less 
than  90°,  and  the  lower  if  it  is  greater  than  90°. 

Since  the  projections  of  the  areal  velocity  upon  the  three  funda- 
mental planes  are  constants  (viz.,  Jai,  fa2,  and  ^a3),  the  areal  veloc- 
ity in  the  plane  of  the  orbit  is  also  constant.  Let  this  constant 
be  represented  by  Jcij  then 


(14)  ci  =  Vai2  +  a22  +  a32, 

where  the  positive  value  of  the  square  root  is  taken.  On  solving 
(13)  and  (14)  for  a1}  a2,  and  a3,  it  is  found  that 

a\  =  +  Ci  cos  i, 

(15)  -  a2  =  =*=  Ci  sin  i  sin  ft, 

.  ct3  =  =F  Ci  sin  i  cos  ft , 

where  the  upper  or  lower  signs  are  to  be  taken  in  the  last  two 
equations  according  as  i  is  less  than  or  greater  than  90°;  that  is, 
according  as  ai  is  positive  or  negative.  With  this  understanding 
equations  (15)  uniquely  determine  i  and  ft,  which  uniquely 
determine  the  position  of  the  plane  of  the  orbit. 

87.  Problem  in  the  Plane.  Since  the  orbit  lies  in  a  known 
plane,  the  coordinate  axes  may  be  chosen  so  that  the  x  and  ?/-axes 
lie  in  this  plane.  If  the  coordinates  are  represented  by  x  and  y 
as  before,  the  differential  equations  of  motion  are 

dzx 


The  problem  is  now  of  the  fourth  order  instead  of  the  sixth  as 
it  was  in  (9),  having  been  reduced  by  means  of  the  integrals  (10). 
It  will  be  observed  that,  since  the  position  of  the  plane  is  defined 
by  the  two  elements  ft  and  i,  or  by  the  ratios  of  ai,  a2,  and  a3  in 
(11),  only  two  of  the  arbitrary  constants  were  involved  in  the 
reduction.  This  problem  might  be  solved  by  deriving  the  differ- 


87]  PROBLEM  IN  THE  PLANE.  147 

ential  equation  of  the  orbit  as  in  Art.  54  and  integrating  as  in 
Art.  62,  the  last  integral  being  derived  from  the  integral  of  areas; 
but,  it  is  preferable  to  obtain  the  results  directly  by  the  method 
which  is  usually  employed  in  Celestial  Mechanics. 

Equations  (16)  give 

d?y          d*x  _ 
XW~ydt*  ~ 

The  integral  of  this  equation  is 

dy         dx 

•*•"*?"/" 

which  becomes  in  polar  coordinates 

on  *S-* 

Let  A  represent  the  area  swept  over  by  the  radius  vector  r;  then 

o  dA        *de 
*-&  =  +*  =  *' 
whence 

(18)  2A=Cl*  +  c2, 

from  which  it  follows  that  the  areas  swept  over  by  the  radius 
vector  are  proportional  to  the  times  in  which  they  are  described. 

On  multiplying  (16)  by  2  -=-  and  2  -j-  respectively,  and  adding, 
the  result  is 

2k2M  dr 


^xdx          ^ydy_  k?M  /    dx          dy\_ 

Z  dt2   dt  ^    '  dt2  dt  =  r3     \X  dt  ±y  dt  )  ~~  r2     dt  ' 

The  integral  of  this  equation  is 


This  equation,  which  involves  only  the  square  of  the  velocity 
and  the  distance,  is  known  as  the  vis  viva  integral  (Art.  52).  On 
transforming  the  left  member  to  polar  coordinates,  this  equation 
becomes 


dd\z 
U)   = 


But- 

dr  dr  dO 

Tt=Tedt 

therefore 


148  ELEMENTS   AND    CONSTANTS    OF   INTEGRATION.  [88 

-  ™M.  _L  c 

r 
On  eliminating  -j-  by  means  of  (17),  this  equation  gives 

de  = 


V-  d2  +  2/cWr  +  c3r2' 
which  may  be  written  in  the  form 


(20)  de  = 


Let  B2  and  M  be  defined  by 


in  which  B2  must  be  positive  for  a  real  orbit;  then  (20)  becomes 

—  du 

dB  = 


,  B2  -  u2 
The  integral  of  this  equation  is 

6  =  cos-1  ^  +  c4. 

On  changing  from  u,  B,  and  C4  to  r  and  the  original  constants,  it  is 
found  that 


(21)  r  - 


k2M         I 

-^ -V/ 


Cl         V*^!?-' 


which  is  the  polar  equation  of  a  conic  section  with  the  origin  at 
one  of  its  foci. 

88.  The  Elements  in  Terms  of  the  Constants  of  Integration. 

The  node  and  inclination  are  expressed  in  terms  of  the  constants 
of  integration  by  (15). 

The  ordinary  equation  of  a  conic  section  with  the  origin  at  the 
right-hand  focus  is 

P 


r  = 


1  +  e  cos  (6  -  co) ' 


89] 


PROPERTIES    OF   THE    MOTION. 


149 


where  p  is  the  semi-parameter,  and  o>  is  the  angle  between  the 
polar  axis  and  the  major  axis  of  the  conic.  On  comparing  this 
equation  with  (21),  it  is  found  that 


(22) 


P  = 


Ci2C3 


e2  =  1  + 
co  =  04  —  T; 
d  =  k^Mp, 


P 


M. 


When  e2  <  1,  the  orbit  is  an  ellipse  and  p  =  a(l  —  e2),  where 
a  is  the  major  semi-axis;  when  e2  =  1,  the  orbit  is  a  parabola  and 
p  =  2q,  where  q  is  the  distance  from  the  origin  to  the  vertex  of 
the  parabola;  and  when  e2  >  1,  the  orbit  is  an  hyperbola  and 
p  =  a(e2  -  1). 

Let  AQ  represent  the  area  described  at  the  time  the  body  passes 
perihelion;*  then  the  time  of  perihelion  passage  is  found  from 
equation  (18)  to  be 

2A0  -  c2 


(23) 


T  = 


Ci 


This  completes  the  determination  of  the  elements  in  terms  of 
the  constants  of  integration.  They  are  denned  in  terms  of  the 
initial  coordinates  and  components  of  velocity  by  the  equations 
where  they  first  occur,  viz.,  (10),  (17),  (18),  (19),  and  (21). 

89.  Properties  of  the  Motion.  Suppose  the  orbit  is  an  ellipse. 
Then,  when  the  values  of  the  constants  of  integration  given  in 
(22)  are  substituted  in  (17)  and  (19),  these  equations  become 


(24) 


where  V  is  the  speed  in  the  orbit  at  the  distance  r  from  the  origin. 
When  the  orbit  is  a  circle,  r  =  a  and 

*  Unless  Wa  is  specified  to  be  some  body  other  than  the  sun  the  nearest  apse 
will  be  called  the  perihelion  point. 


150 


PROPERTIES   OF   THE   MOTION. 


[89 


When  the  orbit  is  a  parabola,  a  =  oo  and 


V     ' 


Therefore,  at  a  given  distance  from  the  origin  the  ratio  of  the 
speed  in  a  parabolic  orbit  to  that  in  a  circular  orbit  is 

(25)  7p:7e  =  V2:l. 

Thus,  in  the  motion  of  comets  around  the  sun  they  cross  the 
planets'  orbits  with  velocities  about  1.414  times  those  with  which 
the  respective  planets  move. 

The  speed  that  a  body  will  acquire  in  falling  from  the  distance 
s  to  the  distance  r  toward  the  center  of  force  k2M  is  given  by 
(see  Art.  35) 

V2  =  2kzM  (---}. 
\r      sj 

If  s  is  determined  by  the  condition  that  this  shall  equal  the  speed 
in  the  orbit,  it  is  found,  after  equating  the  right  member  of  this 


Fig.  26. 

expression  to  the  right  member  of  the  second  of  (24),  that  s  =  2a 
and 

(26)  F* 

Therefore,  the  speed  of  a  body  moving  in  an  ellipse  is  at  every 


89]  PROPERTIES   OF   THE   MOTION.  151 

point  equal  to  that  which  it  would  acquire  in  falling  from  the  circum- 
ference of  a  circle,  with  center  at  the  origin  and  radius  equal  to  the 
major  axis  of  the  conicf  to  the  ellipse. 

The  speed  at  P  in  the  ellipse  is  equal  to  that  which  would  be 
acquired  in  falling  from  P'  to  P. 

Equation  (26)  gives  an  interesting  conclusion  about  the  possible 
motion  of  m\  on  the  basis  of  this  equation  alone,  and  without 
making  any  use  of  the  detailed  properties  of  motion  in  a  conic 
section.  Since  the  left  member  is  necessarily  positive  (or  zero) 
r  can  take  only  such  values  that  the  right  member  shall  be  positive 
(or  zero).  Consequently  r  ^  2a  in  all  the  motion  whatever  it 
may  be.  This  result  is  trivial  in  this  simple  case  in  which  all 
the  circumstances  of  motion  are  fully  known,  but  the  corresponding 
discussion  in  the  Problem  of  Three  Bodies  (Chap,  vm.)  gives  valu- 
able information  which  has  not  been  otherwise  obtained. 

Consider  the  second  equation  of  (24)  and  suppose  the  body 
mi  is  projected  from  a  point  which  is  distant  r  from  the  body  w2. 
It  follows  at  once  that  the  major  axis  of  the  conic  depends  upon  the 
initial  distance  from  the  origin  and  the  initial  speed,  but  not  upon 


the  direction  of  projection.     If  V2  <  -  =  U2,  which  is  the  veloc- 

ity the  body  mi  would  acquire  in  falling  from  infinity,  a  is  positive 
and  the  orbit  is  an  ellipse;  if  V2  =  U2,  a  is  infinite  and  the  orbit 
is  a  parabola  ;  if  V2  >  U2,  a  is  negative  and  the  orbit  is  an  hyperbola. 
Let  ti  and  tz  be  two  epochs,  and  AI  and  Az  the  corresponding 
values  of  the  area  described  by  the  radius  vector.  Then  equation 

(18)  gives 

2(A2  -  Ai)  =  (t2  -  Zi)ci. 

Suppose  tz  —  h  =  P,  the  period  of  revolution;  then  2(A2  —  AI) 
equals  twice  the  area  of  the'  ellipse,  which  equals  2irab.  The 
expression  for  the  period,  found  by  substituting  the  value  of  Ci 
given  in  (22)  and  solving,  is 

' 


From  this  equation  it  follows  that  the  period  is  independent  of 
every  element  except  the  major  axis;  or,  because  of  (26),  the  period 
depends  only  upon  the  initial  distance  from  the  origin  and  the 
initial  speed,  and  not  upon  the  direction  of  projection.  The 
major  semi-axis  will  be  called  the  mean  distance,  although  it  must 
be  understood  that  it  is  not  the  average  distance  when  the  time  is 


152 


PROPERTIES   OF   THE   MOTION. 


[89 


used  as  the  independent  variable.  (See  Probs.  4  and  5,  p.  154.) 
The  three  orbits  drawn  in  Fig.  27  have  the  same  length  of 
major  axis  and  are  consequently  described  in  the  same  time. 
The  speed  of  projection  from  A  is  the  same  in  each  case,  the 
differences  in  the  shapes  and  positions  resulting  from  the  different 
directions  of  projection. 


Fig.  27. 

If  the  two  systems  mi,  mz,  and  mz,  ms  are  considered,  and  the 
ratio  of  their  periods  is  taken,  it  is  found  that 


P2i, 


_  U*  1,  2 
3,  2          a33,  2 


M 


3,  2 


1,2 


If  the  two  systems  are  composed  of  the  sun  and  two  planets 
respectively,  then  MI,  2  and  Ma, 2  are  very  nearly  equal  because 
the  masses  of  the  planets  are  exceedingly  small  compared  to  that 
of  the  sun.  Therefore,  this  equation  becomes  very  nearly 

P\  2      a\  2 


or,  the  squares  of  the  periodic  times  of  the  planets  are  proportional  to 
the  cubes  of  their  mean  distances.     This  is  Kepler's  third  law. 

It  is  to  be  observed  that,  in  taking  the  ratios  of  the  periods,  it 
was  assumed  that  k  has  the  same  value  for  the  different  planets; 
that  is,  that  the  sun's  acceleration  of  the  two  planets  would  be 
the  same  at  unit  distance.  On  the  other  hand,  it  follows  from  the 
last  equation,  which  Kepler  established  directly  by  observations, 
that  k  has  the  same  value  for  the  various  planets.  This  means 
that  the  force  of  gravitation  between  the  sun  and  the  several 


90]  SELECTION   OF   UNITS.  153 

planets  is  proportional  to  their  respective  masses,  as  measured 
by  their  inertias.  This  result  is  not  self-evident  for  the  force  of 
gravitation  conceivably  might  depend  upon  the  chemical  con- 
stitution or  physical  condition  of  a  body,  just  as  chemical  affinity, 
magnetism  and  all  other  known  forces  depend  upon  one  or  both 
of  these  things.  In  fact,  it  is  remarkable  that  gravitation  is 
proportional  to  inertia  and  independent  of  everything  else. 

90.  Selection  of  Units  and  the  Determination  of  the  Constant  k. 

When  the  units  of  time,  mass,  and  distance  are  chosen  k  can  be 
determined  from  (27).  It  is  evident  that  they  can  all  be  taken 
arbitrarily,  but  it  will  be  convenient  to  employ  those  units  in 
which  astronomical  problems  are  most  frequently  treated.  The 
mean  solar  day  will  be  taken  as  the  unit  of  time;  the  mass  of  the 
sun  will  be  taken  as  the  unit  of  mass;  and  the  major  semi-axis  of 
the  earth's  orbit  will  be  taken  as  the  unit  of  distance.  When  these 
units  are  employed  the  k  determined  by  them  is  called  the  Gaussian 
constant,  having  been  defined  in  this  way  by  Gauss  in  the  Theoria 
Motus,  Art.  1. 

Let  ra2  represent  the  mass  of  the  sun  and  mi  that  of  the  earth 
and  moon  together;  then  it  has  been  found  from  observation  that 
in  these  units 


(28) 


mi  " 


354710  "354710' 
.  P  =  365.2563835. 

On  substituting  these  numbers  in  (27),  it  is  found  that 

f        k  =  —  .  2?r         =  0.01720209895, 

(29)  PVl  +  m! 

I  log  k  =  8.2355814414  -  10. 

Since  mi  is  very  small  k  =  -p-  nearly,  and  is,  therefore,  nearly 
the  mean  daily  motion  of  the  earth  in  its  orbit,  or  about  -fa.     The 

mean  daily  motion  of  a  planet  whose  mass  is  m^  is  -=-  ,  and  is 

•*  » 

usually  designated  by  nt-.     This  is  found  from  (27)  to  be 

(30)  «' 

The  period  of  the  earth's  revolution  around  the  sun  and  its 
mean  distance  were  not  known  with  perfect  exactness  at  the 


154  PROBLEMS. 

time  of  Gauss,  nor  are  they  yet,  and  it  is  clear  that  the  value  of 
k  varies  with  the  different  determinations  of  these  quantities. 
If  astronomers  held  strictly  to  the  definitions  of  the  units  given 
above  it  would  be  necessary  to  recompute  those  tables  which 
depend  upon  k  every  time  an  improvement  in  the  values  of  the 
constants  is  made.  These  inconveniences  are  avoided  by  keeping 
the  numerical  value  of  k  that  which  Gauss  determined,  and 
choosing  the  unit  of  distance  so  that  (27)  will  always  be  fulfilled. 
If  the  mean  distance  between  two  bodies  is  taken  as  the  unit  of 
distance  and  the  sum  of  their  masses  as  the  unit  of  mass,  and  if  the 
unit  of  time  is  taken  so  that  k  equals  unity,  then  the  units  form 
what  is  called  a  canonical  system.  Since  M  =  1  and  k2  =  1  in 
this  system,  and  from  (30)  n  =  1,  the  equations  become  some- 
what simplified  and  are  advantageous  in  purely  theoretical 
investigations. 


XIV.     PROBLEMS. 

1.  Find  the  differential  equations  for  the  problem  of  the  relative  motion  of 
two  bodies  in  polar  coordinates. 

Ans.  T—  —  r  (  —  )    —  -j- 

dt 2  \  at  /  i*  at 

2.  Integrate  the  equations  of  problem  1  and  interpret  the  constants  of 
integration. 

3.  The  earth  moves  in  its  orbit,  which  may  be  assumed  to  be  circular,  with 
a  speed  of  18.5  miles  per  second.     Suppose  the  meteors  approach  the  sun  in 
parabolas;  between  what  limits  will  be  their  relative  speed  when  they  strike 
into  the  earth's  atmosphere? 

Ans.  7.66  to  44.66  miles  per  second.  (The  Nov.  14  meteors  meet  the 
earth  and  have  a  relative  speed  near  the  upper  limit;  the  Nov.  27  meteors 
overtake  the  earth  and  have  a  relative  speed  near  the  lower  limit.) 

4.  Find  the  average  length  of  the  radius  vector  of  an  ellipse  in  terms  of 
a  and  e,  taking  the  time  as  the  independent  variable. 

j*rdt 
Ans.  Average  r  =  — —  = 


5.  Find  the  average  length  of  the  radius  vector  of  an  ellipse,  taking  the 
angle  as  the  independent  variable. 

frdd      2*aVT=l* 
Ans.  Average  r  =  ~= —  =  —  -  =  b. 

fa 


91]  POSITION    IN    PARABOLIC    ORBITS.  155 

6.  Prove  that  the  amount  of  heat  received  from  the  sun  by  the  planets 
per  unit  area  is  on  the  average  proportional  to  the  reciprocals  of  the  products 
of  the  major  and  miner  axes  of  their  orbits.     For  a  fixed  major  axis  how  does 
the  total  amount  of  heat  received  in  a  revolution  depend  upon  the  eccentricity 
of  the  orbit? 

7.  If  particles  are  projected  from  a  given  point  with  a  given  velocity  but 
in  different  directions,  find  the  locus  of  (a)  perihelion  points;  (6)  aphelion 
points;  (c)  centers  of  ellipses;  (d)  ends  of  minor  axes. 

8.  If  particles  are  projected  from  a  given  point  in  a  given  direction  but 
with  different  speeds,  find  the  loci  of  the  same  points  as  in  problem  7,  and 
express  the  coordinates  of  these  points  in  terms  of  the  initial  values  of  the 
coordinates  and  the  components  of  velocity. 

9.  Suppose  a  comet  moving  in  a  parabolic  orbit  with  perihelion  distance  q : 
collides  with  and  combines  with  an  equal  mass  which  is  at  rest  before  the 
collision.    Find  the  eccentricity  and  the  perihelion  distance  of  the  orbit  of 
the  combined  mass. 

10.  Suppose  the  mass  of  Jupiter  is  1/1047  when  expressed  in  terms  of  the 
mass  of  the  sun,  and  that  its  mean  distance  from  the  sun  is  483,300,000  miles 
(the  mean  distance  from  the  earth  to  the  sun  is  .92,900,000  miles).     Find 
Jupiter's  period  of  revolution  around  the  sun,  and  the  size  of  the  orbit  which 
the  sun  describes  with  respect  to  the  center  of  gravity  of  itself  and  Jupiter. 

91.  Position  in  Parabolic  Orbits.  Having  found  the  curves  in 
which  the  bodies  move,  it  remains  to  find  their  positions  in  their 
orbits  at  any  given  epoch.  The  case  of  the  parabolic  orbit  being 
the  simplest  will  be  considered  first,  and  it  will  be  supposed,  to 
fix  the  ideas,  that  the  motion  is  that  of  a  comet  with  respect  to 
the  sun.  Since  the  masses  of  the  comets  are  negligible,  M  =  1 
and  equation  (17)  becomes 

When  the  polar  angle  in  the  orbit  is  counted  from  the  vertex  of 
the  parabola  it  is  denoted  by  v,  and  is  called  the  true  anomaly. 
Then 

'd^dr 

r  =  -  -r-Z =  q  sec2 » . 

1  +  cos  v  2 


Hence,  equation  (31)  gives 
| 


-  dt  =  sec4  —  dv  —  (  sec2  ^  +  sec2  —  tan2  —  1  dv. 


156  POSITION   IN   PARABOLIC    ORBITS.  [91 

The  integral  of  this  expression  is 
(32) 


where  T  is  the  time  of  perihelion  passage.     This  is  a  cubic  equation 

in  tan  ~  .     On  taking  the  right  member  to  the  left  side  it  is  seen 
& 

that  for  t  —  T  >  0,  the  function  is  negative  when  v  =  0,  and  that 
it  increases  continually  with  v  until  it  equals  infinity  for  v  =  180°. 

Therefore,  there  is  but  one  real  solution  of  (32)  for  tan  —  ,  and  it 

ft 

is  positive.     For  t  —  T  <  0  it  is  seen  in  a  similar  manner  that 
there  is  one  real  negative  solution. 
Equation  (32)  may  be  written 


Tables  have  been  constructed  giving  the  value  of  the  right  member 
of  this  equation  for  different  values  of  v.  From  these  tables  v  can 
be  found  by  interpolation  when  t  —  T  is  given;  or,  conversely, 
t  —  T  can  be  found  when  v  is  given.  These  tables  are  known  as 
Barker's,  and  are  VI.  in  Watson's  Theoretical  Astronomy,  and  IV. 
in  Oppolzer's  Bahnbestimmung.  * 

In  order  to  find  the  direct  solution  of  the  cubic  equation  let 


whence 


tan—  =  2  cot  2w  =  cot  w  —  tan  w; 


tan3  —  =  —  3  tan  —  +  cot3  w  —  tan3  w. 


This  substitution  reduces  (32)  to 

3k(t  -  T) 
cot3  w  —  tan3  w  = 

Let 


whence 

cots=3fc(2^«7')- 
Therefore  the  formulas  for  the  computation  of  tan  ^  are,  in  the 

*  In  Oppolzer's  Bahnbestimmung  the  factor  75  is  not  introduced. 


92]  EULER^S   EQUATION. 

order  of  their  application, 


157 


(33) 


COt    S 


cot  w 


3k(t  -  T) 


3/        ,    S 

=  \cot2' 


tan  —  =  2  cot  2w. 


After  v  has  been  found  r  is  determined  by  the  polar  equation  of 


the  parabola,  r  = 


cos  v 


q  sec2  - 


2' 


92.  Equation  involving  Two  Radii  and  their  Chord.  Euler's 
Equation.  Consider  the  positions  of  the  comet  at  the  instants 
ti  and  Z2.  Let  the  corresponding  radii  be  r\  and  r2,  and  the  chord 
joining  their  extremities  s.  Let  the  corresponding  true  anomalies 
be  Vi  and  vz.  Then  it  follows  that 

k(ti  -  T)  »i  . 

J 


-  T} 


The  difference  of  these  equations  is 


01, 


(M) 

The  equation  for  the  chord  is 

s2  =  ri2  +  r22  —  2rir2  cos 


From  this  equation  it  is  found  that 
(35)   2V^c 


s)(ri 


-  s). 


The  +  sign  is  to  be  taken  before  the  radical  if  vz  —  Vi  <  IT,  and 
the  —  sign  if  v2  —  Vi  >  TT. 


158  POSITION   IN   ELLIPTIC    ORBITS.  [93 

It  follows  from  the  polar  equation  of  the  parabola  that 

,        r2  =  gsec2^. 
These  expressions  for  ri  and  r2,  substituted  in  (35),  give 


(36)      1  +  tan      tan      =  ±      r'      r' 


It  also  follows  from  the  expressions  for  r\  and  r2  that 

ri  +  r2  =  «  (2  +  tan2  £  + 
The  last  two  equations  give 


(ri  +  r2  +  s)  +  (ri  +  r2  -  s)  =F  2  V(ri  +  r2  +  s)(ri  +  r2  -  s) 

.(tanl-tan 

whence 


/Q7N         +  yy-j-  s  =F      ri  +  r2  -  s         v2          vi 

---  =tan--   tan-. 


Equation  (34)  becomes,  as  a  consequence  of  (36)  and  (37), 
(38)  6fc(*2  -  «i)  =  (ri  +  r2  +  s)*  ^  (ri  +  r2  -  s)i. 

This  equation  is  remarkable  in  that  it  does  not  involve  q.  It  was 
discovered  by  Euler  and  bears  his  name.  It  is  of  the  first  im- 
portance in  some  methods  of  determining  the  elements  of  a  para- 
bolic orbit  from  geocentric  observations. 

There  is  a  corresponding  equation,  due  to  Lambert,  for  elliptic 
orbits.  The  right  member  is  developed  as  a  power  series  in  I/a, 
the  first  term  constituting  the  right  member  of  Euler's  equation. 

93.  Position  in  Elliptic  Orbits.  The  integral  of  areas  and  the 
vis  viva  integral  are  respectively 


dv 


/dr\ 
(dt) 


The  result  of  eliminating  -=-  from  the  second  of  these  equations 


94]  GEOMETRICAL   DERIVATION   OF   KEPLER'  S   EQUATION.          159 

by  means  of  the  first  is 

(39) 


Let  n  represent  the  mean  angular  motion  of  the  body  in  its  orbit  ; 
then  _ 

2ir      k-^l+m 
n=T  =       ~^~ 

On  introducing  n  in  (39)  and  solving,  it  is  found  that 
(40)  ndt=T- 


(41) 


a  Va2e2  -  (a  -  r)2 

In  order  to  normalize  the  integral  which  appears  in  the  right 
member  of  (40),  let  the  auxiliary  E  be  introduced  by  the  equation 

a  —  r  =  ae  cos  E,     whence 

r  =  a(l  —  e  cosE). 
This  angle  E  is  called  the  eccentric  anomaly.    Then  (40)  becomes 

ndt  =  (1  -  ecosE)dE, 
the  integral  of  which  is 

n(t-  T)  =  E  -esmE. 

The  quantity  n(t  —  T)  is  the  angle  which  would  have  been  de- 
scribed by  the  radius  vector  if  it  had  moved  uniformly  with  the 
average  rate.  It  is  usually  denoted  by  M  and  is  called  the  mean 
anomaly.  Therefore 

(42)  n(t  -  T)  =  M  =  E  -  e  sin  E. 

The  M  can  at  once  be  found  when  (t  —  T)  is  given,  after  which 
equation  (42)  must  be  solved  for  E.  Then  r  and  v  can  be  found 
from  (41)  and  the  polar  equation  of  the  ellipse.  Equation  (42), 
known  as  Kepler's  equation,  is  transcendental  in  E,  and  the  solution 
for  this  quantity  cannot  be  expressed  in  a  finite  number  of  terms. 
Since  it  is  very  desirable  to  have  the  solution  as  short  as  possible 
astronomers  have  devoted  much  attention  to  this  equation,  and 
several  hundred  methods  of  solving  it  have  been  discovered. 

94.  Geometrical  Derivation  of  Kepler's  Equation.  Construct 
the  ellipse  in  which  the  body  moves,  and  also  its  auxiliary  circle 
AQB.  The  angle  AFP  equals  the  true  anomaly,  v;  the  angle 


160 


SOLUTION   OF   KEPLER  S   EQUATION. 


[95 


ACQ  will  be  defined  as  the  eccentric  anomaly,  E,  and  it  will  be 
shown  that  the  relation  between  M  and  E  is  given  by  Kepler's 
equation. 

.3 


Fig.  28. 

From  the  law  of  areas  and  the  properties  of  the  auxiliary  circle, 
it  follows  that 

M  __  area  AFP  =  area  AFQ 
2ir      area  ellipse      area  circle ' 

2  Tji          n 

Area  AFQ  =  area  ACQ  -  area  FCQ  =  -= ae  sin  E. 

2         2i 

Therefore 

M_  _  a2  (E  -  e  sin  E) 
27r  ~  2  ~        Tra2        '    : 


or, 


M  =  E  —  e  sin  E, 


FP  =  r 


e  cos 


FD   =  al-  e  cos 


which  is  the  definition  of  the  eccentric  anomaly  given  in  (41). 

95.  Solution  of  Kepler's  Equation.  It  will  be  shown  first  that 
Kepler's  equation  always  has  one,  and  only  one,  real  solution  for 
every  value  of  M  and  for  every  e  such  that  0  ^  e  <  1.  Write 
the  equation  in  the  form 


4>(E)  =  E  -  e  sin  E  -  M  =  0. 

Suppose  M  has  some  given  value  between  rnr  and  (n  +  l)?r, 
where  n  is  any  integer;  then  there  is  but  one  real  value  of  E  satis- 
fying this  equation,  and  it  lies  between  mr  and  (n  +  I)TT.  For, 
the  function  <f>(E)  when  E  =  mr  is 

<j>(nir)  =  mr  —  M  <  0. 


95]  SOLUTION  OF  KEPLER'S  EQUATION.  161 

And  4>(E)  when  E  —  (n  +  I)TT  is 

0[(w  +  !)TT]  =  (n  +  I)TT  -  M  >  0. 

Consequently  there  is  an  odd  number  of  real  solutions  for  E  which 
lie  between  mr  and  (n  +  I)TT.  But  the  derivative 

4>'(E)  =  1  -  e  cos  E 

is  always  positive;  therefore  0(#)  increases  continually  with  E 
and  takes  the  value  zero  but  once. 

A  convenient  method  of  practically  solving  the  equation  is  by 
means  of  an  expansion  due  to  Lagrange.  Suppose  z  is  defined  as 
a  function  of  w  by  the  equation 

(43)  Z  =  W  +  00(2), 

where  a  is  a  parameter.  Lagrange  has  shown  that  any  function 
of  z  can  be  expressed  in  a  power  series  in  a,  which  converges  for 
sufficiently  small  values  of  a,  of  the  form* 

(  F(z)  =  F(w)      *          ™   N    '     a*      d 
(44) 


This  expansion  can  be  applied  to  the  solution  of  Kepler's  equation 

by  writing  it 

E  =  M  +  e  sin  E, 

which  is  of  the  same  form  as  (43).  The  expansion  of  E  as  a  series 
in  e  can  be  taken  from  (44)  by  putting  F(z)  =  E,  0(z)  =  sinJ£, 
w  =  M,  and  a  =  e.  The  result  is 

«•  fc*^ 

(45)  E  =  M  +  r  sinM  +  - — ^sin2M  +  •••. 

1  1  •  Z 

All  the  terms  on  the  right  except  the  first  are  expressed  in  radians 
and  must  be  reduced  to  degrees  by  multiplying  each  of  them  by 
the  number  of  degrees  in  a  radian.  The  higher  terms  are  con- 
siderably more  complicated  than  those  written,  and  the  work  of 
computing  them  increases  very  rapidly.  In  the  planetary  and 
satellite  orbits  the  eccentricity  is  very  small,  and  the  series  (45) 
converges  with  great  rapidity,  the  first  three  terms  giving  quite 
an  accurate  value  of  E. 

*  Williamson's  Diff.  Calc.,  p.  151. 
12 


162  DIFFERENTIAL   CORRECTIONS.  [96 

96.  Differential  Corrections.  A  method  will  now  be  explained 
in  one  of  its  simplest  applications,  which  is  of  great  importance 
in  many  astronomical  problems.  Suppose  an  approximate  value 
of  E  is  determined  by  the  first  three  terms  of  (45).  Call  it  E0', 
it  is  required  to  find  the  correct  value  of  E. 

Kepler's  equation  gives 

MQ  =  EQ  —  e  sin  EQ. 

For  a  particular  value  of  M,  viz.,  M0,  the  corresponding  value  of 
E,  viz.,  EQ,  is  known.  It  is  required  to  find  the  value  of  E  corre- 
sponding to  M,  which  differs  only  a  little  from  M0.  The  angle  M 
is  a  function  of  E  and  may  be  written 


M  =  MQ  +  AM0  =  f(EQ  +  A#0). 

On  expanding  the  right  member  by  Taylor's  formula,  this  equation 
becomes 

M  =  Mo  +  AM0  =  f(EQ)  +  /'(#o)A#0  +  •  •  •• 

By  the  definitions  of  the  quantities,  MQ  —  f(EQ)-  therefore  this 
equation  becomes 

(46)    M  -  MQ=  f(Eo)AEQ  +  -  •  •  =  (1  -  e  cos  #o)A#o  +  •  •  •  . 

Since  &E0  is  very  small  the  squares  and  higher  powers  may  be 
neglected,  *  and  then  equation  (46)  gives  for  the  correction  to  be 
applied  to  EQ 


(47) 


1  —  e  cos 


With  the  more  nearly  correct  value  of  E,  EI  =  EQ  +  AE0,  and 
MI  can  be  computed  from  Kepler's  equation,  and  a  second  correc- 
tion will  be 

A#   =     M-M, 

I  —  e  cos  EI  ' 

This  process  can  be  repeated  until  the  value  of  E  is  found  as  near 
as  may  be  desired.f     In  the  planetary  orbits  two  applications  of 

*  If  the  higher  terms  in  AEo  were  not  neglected  AE0  could  be  expressed  as  a 
power  series  in  M  —  M0,  of  which  the  first  term  would  be  the  right  member 
of  (47). 

t  For  the  proof  of  the  convergence  of  a  similar,  but  somewhat  more  laborious, 
process  see  Appell's  Mecaniqtte  vol.  i.,  p.  391. 


97] 


GRAPHICAL   SOLUTION   OF   KEPLER'S  EQUATION. 


163 


the  formulas  will  nearly  always  give  results  which  are  sufficiently 
accurate,  and  usually  one  correction  will  suffice. 

97.  Graphical  Solution  of  Kepler's  Equation.  When  the 
eccentricity  is  more  than  0.2  the  method  of  solving  Kepler's 
equation  given  above  is  laborious  because  the  first  approximation 
will  be  very  inexact.  These  high  eccentricities  occur  in  binary 
star  and  comet  orbits,  and  are  sometimes  even  so  great  as  0.9. 
In  the  case  of  binary  star  orbits  it  is  usually  sufficient  to  have  a 


y 

90 
80 
70 
60 
50 
W 
SO 
20 
10 
0 

Axis 

zp       40 

60 

80 

JOO 

120 

140 

160    ,     180          2<jO 

| 

J. 

^-j 

06 

e 

E-A&s^s 

/ 

, 

/ 

/ 

^. 

-if- 

iXx- 

-^ 

^ 

h^ 

L 

*W 

J, 

0 

^y 

170 

L 

0 

1< 

0 

. 

^ 

V 

. 

5 

Or 

E 

r 

x 

2 

X 

/ 

s 

/ 

N 

/ 

\ 

/ 

2 

\ 

/ 

\ 

/ 

' 

\ 

/ 

7 

\ 

/ 

j 

\ 

2 

\ 

T  20° 

*~  l£°M      60° 

^  SOB       J00°        1^0°    .    lk)° 

Fig.  29. 

•jft" 

3^0°    '    ^OV 

solution  to  within  a  tenth  of  one  degree.     In  this  work  a  rapid 
graphical  method  is  of  great  practical  value. 
Consider  Kepler's  equation 

E  -  e  sin  E  -  M  =  0, 

where  M  is  given  and  E  is  required.  Take  a  rectangular  system 
of  axes  and  construct  the  sine  curve  and  the  straight  line  whose 
equations  are 

y  =  sin  E, 


The  abscissa  of  their  point  of  intersection  is  the  value  of  E  satis- 
fying the  equation;*  for,  eliminating  y,  Kepler's  equation  results. 
The  first  curve  is  the  familiar  sine  curve  which  can  be  constructed 


*  Due  to  J.  J.  Waterson,  Monthly  Notices,  1849-50,  p.  169. 


164 


RECAPITULATION    OF   FORMULAS. 


[98 


once  for  all;  the  second  is  a  straight  line  making  with  the  E-axis 
an  angle  whose  tangent  is  l/e.  Instead  of  drawing  the  straight 
line  a  straight-edge  can  be  laid  down  making  the  proper  slope 
with  the  axis.  To  facilitate  the  determination  of  its  position 
construct  a  line  with  the  degrees  marked  on  it  at  an  altitude  of 
100;*  then  place  the  bottom  of  the  straight-edge  at  M  and  the 
top  at  M  +  lOOe,  and  it  follows  that  it  will  have  the  proper  slope. 
If  M  is  so  near  180°  that  the  straight-edge  runs  off  from  the 
diagram,  the  top  can  be  placed  at  M  +  50e  on  the  50-line.  As  M 
becomes  very  near  180°  the  mean  and  eccentric  anomalies  become 
very  nearly  equal,  exactly  coinciding  at  M  =  180°. 

98.  Recapitulation  of  Formulas.  The  equations  for  the  com- 
putation of  the  polar  coordinates,  when,  the  time  is  given,  will 
now  be  given  in  the  order  in  which  they  are  used. 


n 


(48) 


EQ  =  M  +  e  sin  M  +  —  sin  2M, 

MQ  =  EQ  —  e  sin  EQ, 

Ag°  =  l-e"co^k' 

#1  =  #o  +  A#o, 

r  =  a(l  —  e  cos  E)  = 


whence 


(49) 


cos  v  = 

sin  v  = 

1  +  cos  v  = 

1  —  cos  v  = 


1  +  e  cos  z; ' 

cos  E  —  e 
1  -  e  cos  # ' 

Vl  —  e2  sin  # 
1  -  e  cos  #   ' 

(1  -e)(l  +  cosE) 

1—6  COS  # 

(1  +e)(l  -  cosE') 


1  —  e  cos  E 
*  This  device  is  due  to  C.  A.  Young. 


99]  DEVELOPMENT   OF   E  IN   SERIES.  165 

The  square  root  of  the  quotient  of  the  last  two  equations  gives  a 
very  convenient  formula  for  the  computation  of  v,  viz., 

(50) 

The  last  equation  of  (48)  and  equation  (50)  give  the  polar  co- 
ordinates when  E  is  known. 

99.  The  Development  of  E  in  Series.  The  equations  which 
have  been  given  are  sufficient  to  enable  one  to  compute  the  polar, 
and  consequently  the  rectangular,  coordinates  at  any  epoch; 
yet,  in  some  kinds  of  investigations,  as  in  the  theory  of  perturba- 
tions, it  is  necessary  to  have  the  developments  of  not  only  E,  but 
also  the  polar  coordinates,  carried  so  far  that  the  functions  are 
represented  by  the  series  with  the  desired  degree  of  accuracy. 

The  application  of  Lagrange's  method  of  Art.  95  to  Kepler's 
equation  gives  E  as  a  power  series  in  e  whose  coefficients  are 
functions  of  M.  This  method  has  been  used  to  get  the  first  terms 
of  the  series  and  it  can  be  continued  as  far  as  may  be  desired. 
It  is  very  elegant  in  practice  and  is  subject  only  to  the  difficulty 
of  proving  its  legitimacy.  But  a  direct  treatment  of  Kepler's 
equation  based  on  more  elementary  considerations  is  not  difficult. 
The  solution  of 

M  =  E  —  e  sin  E 

for  E  is  j?r  when  M  —  jir,  where  j  =  0,  1,  2,  •  •  • ,  whatever  value  e 
may  have.  Moreover,  it  has  been  shown  that  when  e  is  less  than 
unity  the  solution  is  unique  for  all  values  of  M.  When  e  =  0  the 
solution  is  E  =  M  for  all  values  of  M .  If  u  is  defined  by  the 
equation 

E  -  M  =  u 

Kepler's  equation  becomes 

(51)  u  =  e  sin  (M  +  u), 

which  defines  u  in  terms  of  M  and  e.  For  every  value  of  M  different 
from  jir,  for  which  the  solution  is  already  known,  the  right  member 
of 


sin  (M  +  w) 

can  be  expanded  as  a  converging  power  series  in  u.     When  this 
series  is  inverted  u  will  be  given  as  a  power  series  in  e  whose 


166  DEVELOPMENT   OF   E  IN   SERIES.  [99 

coefficients  are  functions  of  M.     Since  u  vanishes  with  e,  it  will 
have  the  form 

(52)  u  =  ^  e  +  u2  e1  +  u3  e3  + 

Instead  of  forming  the  series  in  u  and  then  inverting,  it  is 
simpler  to  substitute  (52)  in  (51)  and  to  determine  ui}  u%,  -  -.  •  by 
the  condition  that  the  result  shall  be  an  identity  in  e.  The  result 
of  the  substitution  is 

+  u2  e2  +  Us  &  +  •  •  •  =  e  sin  M  cos  u  +  e  cos  M  sin  u 

TI/T  f         (u\e -\- u<2,€^  • '  *)2      (u\ e  -f-  •••)4 

=  e  sin  Mil =—. H T-; •  •  • 

2  !  4  ! 


+  e  cos  Ml  (HI  e  +  uz  e2  +  •  •  • )  —  -— — — —  -f-  •  • 

On  equating  coefficients  of  corresponding  powers  of  e,  it  is  found 

that 

,i  =  sin  M, 


=  Ui  cos  M  =  jr  sin  2M  , 


cos  Af  =  5  sin  3M  —  -  sin  M, 

O  O 


Some  general  properties  of  the  solutions  easily  follow  from 
these  equations.  It  follows  from  (51)  that  if  for  any  M  =  MQ 
the  solution  for  u,  which  is  known  to  exist  uniquely,  is  u  =-  UQ, 
then  the  solution  for  M  =  MQ  +  2jV  (j  any  integer)  is  also  u  =  UQ. 
Therefore  u  is  a  periodic  function  of  M  with  the  period  2?r.  Since 
this  is  true  for  all  values  of  e,  each  Uj  is  separately  periodic  with  the 
period  2?r.  If  any  M0  and  UQ  satisfy  (51),  then  —  M0  and  —  UQ 
also  satisfy  (51);  therefore  u  is  an  odd  function  of  M  and  the  Uj 
are  sines  of  multiples  of  M.  If  the  sign  of  e  is  changed  and  TT  is 
added  to  M  in  (51),  the  equation  is  unchanged;  therefore  the 
Uj  with  odd  subscripts  involve  only  sines  of  odd  multiples  of  M , 
and  those  with  even  subscripts  only  sines  of  even  multiples  of  M. 

It  will  be  shown  that  the  highest  multiple  of  M  appearing  in 
Uj  is  jM.  The  general  term  of  (53)  is 

Uj  =  sinM  PJ(UI,  u2,  •  •  -,  Uj-i)  +  cos  M  QJ(UI,  uz,  •  •  •,  w/_i), 

where  P3-  and  Qj  are  polynomials  in  u\,  uz,  •  •  • ,  w/_i.  These 
quantities  must  enter  in  such  powers  that  they  are  multiplied 
by  e*-1.  Suppose  the  general  terms  of  the  polynomials  P,  and  Qj 


99]  DEVELOPMENT   OF  E   IN    SERIES.  167 

are,  except  for  numerical  coefficients  which  do  not  enter  into  the 
present  argument,  respectively 


j  =  u1? 


The  exponents  of  u\,  •••,  Uj-\  are  subject  to  the  condition  that 
PJ  and  qj  shall  be  multiplied  by  e*~l.  The  term  um  carries  with  it 
the  factor  em,  and  therefore  u^  carries  the  factor  mn.  Hence  the 
exponents  .of  Ui,  •  •  •  ,  Uj-.\  in  PJ  and  q3-  must  satisfy 


(54) 

'    2k. 

Now  suppose  that  the  highest  multiples  of  M  in  um  is  mM  for 
m  =  1,  •  •  • ,  j  —  1.  It  follows  from  the  properties  of  powers  of  the 
sines  that  the  highest  multiple  in  u^  is  mnM.  Since  the  highest 
multiple  of  the  product  of  two  or  more  sines  is  the  sum  of  their 
highest  multiples,  the  highest  multiples  in  PJ  and  q,  are  respec- 
tively 

which  are  j  —  1  by  (54) .  But  it  follows  from  (53)  that  p}-  is  multi- 
plied by  sinTlf  and  q,  by  cos  M ;  therefore  the  highest  multiple 
appearing  in  u3-  is  jM.  That  is,  u,  has  the  form 

(•     U2k  =  +  42*)  sin  2M  +  •  •  •  +  ogf)  sin  2kM, 

(55) 

"smM  -+•  •'•  +  a&*;V;  sin 


according  as  j  is  even  or  odd. 

It  is  easy  to  develop  a  check  on  the  accuracy  of  the  compu- 
tations.    Since  E  =  M  +  u,  it  follows  that 

M=  1+-^=  l  +^!e  4.^2    ,  +^ie; 

dM  ^aM  ^(9Me+c)Me  ^6 

But  it  follows  from  Kepler's  equation  that 


Suppose  Af  =  0;  therefore  E  =  0  and  for  this  value  of  M 

r^T?  1 


Therefore,  since  the  coefficient  of  e'  in  this  series  is  unity,  for  M 


168 


DEVELOPMENT  OF   E  IN   SERIES. 


[99 


(57) 


dM 


(2k 


These  equations  constitute  a  valuable  check  on  all  the  compu- 
tations. 

It  is  found  from  (56)  that 

-  e  sin  E      dE  -  e  sin  E 


dM2      (I  -  e  cos  E)2  dM      (1  -  e  cos  E)* ' 
-  e  cos  E  3e2  sin2  E 


(I  —  e  cos  J5/)4      (1  —  e  cos  £')5 

For  M  =  0,  the  first  of  these  equations  is  identically  zero,  but  the 
second  one  becomes 


d3E 


-<r^"t 


1-2 


4-5  •••  (n  +  2) 
1-2  •••  (n-  1) 


] 


Then  the  conditions  similar  to  (57)  are 


(58) 


4.5  ...  (2k  +  2) 
1-2  ..-  (2/b  -  1) 


1»2  •  •  •  2A; 

These  equations  constitute  a  check  which  is  independent  of  that 
given  in  (57).  In  a  similar  way  check  formulas  can  be  found 
from  a  consideration  of  all  odd  derivatives  of  E  with  respect  to  M. 

Equations  (57),  (58),  and  similar  ones  for  higher  derivatives 
of  E}  are  linear  in  the  coefficients  af\  which  it  is  desired  to  find; 
consequently,  when  the  number  of  equations  equals  the  number 
of  unknowns,  the  latter  are  uniquely  determined,  at  least  if  the 
determinant  of  the  coefficients  is  not  zero.  It  can  be  shown  that 
the  determinant  is  not  zero. 

For  the  purposes  of  illustration  suppose  k  =  0.  Then  the 
second  equation  of  (57)  gives  a^  =  1,  whence  u\  =  sin  M 


100] 


DEVELOPMENT   OF   T  AND   V   IN    SERIES. 


169 


agreeing  with  the  result  in  (53).  Suppose  k  =  1;  then  the  first 
equation  of  (57)  gives  2a(22)  =  1,  whence  u2  —  J  sin  2M.  As  an 
illustration  involving  both  (57)  and  (58),  suppose  k  =  1  and 
consider  the  second  equations  of  (57)  and  (58).  They  become 
in  this  case 


3a(33)= 


,(3) 


ii5 

1-2' 


whence  a(^  =  —  f ,  a(33)  =  +  f ,  agreeing  with  the  results  given 
in  (53). 

When  the  expansion  is  carried  out  by  the  method  of  Lagrange,    M 
or  by  that  which  has  just  been  explained,  the  value  of  E  to  terms 
of  the  sixth  order  in  e  is  found  to  be 


(59) 


E  =  M  +  e  sin  M  +  |-  sin  2M 

(32sin3M-3sinM) 


3!22 

e4 
4!23 

e5 
5!24 

/j6 


(43  sin  4M  -  4  -  23  sin  2M) 
(54  sin  5M  -  5  •  34  sin  3M  +  10  sin  M) 
—  6  •  45sin4Af  +  15  •  25sin 


100.  The  Development  of  r  and  v  in  Series.  The  value  of  r  in 
terms  of  e  and  M  can  be  obtained  by  the  method  of  Lagrange  by 
letting  F(z)  =  cos#  and  making  use  of  the  last  equation  of  (48). 
This  method  has  the  disadvantage  of  being  rather  laborious. 

It  follows  from  Kepler's  equation  that  * 

BE          e  sin  E 


Therefore 


de       I  —  e  cos  E  ' 
dM=(l  -  e  cosE)dE. 


-dM  =  esmEdE. 
de 


The  method  employed  in  this  Art.  is  due  to  MacMillan. 


170  DEVELOPMENT   OF   T    AND   V   IN    SERIES.  [100 

The  integral  of  this  equation  gives 

f*M  f\~j? 

(60)  e  I       -dM  =  -  ecosE  +  c, 

Jo     de 

which  expresses  —  e  cos  E  in  terms  of  M  very  simply  by  sub- 
stituting in  the  left  member  the  explicit  value  of  E  given  in  (59). 
For  example,  the  first  terms  are 


—  e  cos  E  =  —  c  +  e  I        sin  M  +  e  sin  2M 

+  |e2(3  sin  3M  -  sinM)  +  •  •  -1  dM 

1  3 

=  —  c  —  ecos  M  —  ~e2cos2M  —  -^e3(cos3M  —  cosM)  •  -  • . 

Z  o 

The  last  equation  of  (48)  and  (60)  give  for  r  the  series 

(61)      -  =  1  —  e  cos  E  =  1  —  c  —  e  cos  M  —  -  e2  cos  2M 
a  2i 

It  remains  to  determine  the  constant  c.  Since  r  is  measured 
from  the  focus  of  the  ellipse,  it  follows  that  r  =  a(l  —  e)  at 
M  =  0;  whence 


where  6,-  is  the  coefficient  of  e1'  in  the  series  for  —  e  cos  E  at  M  =  0. 
The  two  sides  of  this  equation  must  be  the  same  for  all  values 
of  e  for  which  (61)  converges;  therefore  c  must  have  the  form 

c  =  c2e2  +  c3e3  +  •  •  •, 

where  C2,  c?,,  •  •  •  are  determined  so  that  the  right  member  will 
contain  no  terms  in  e2,  e3,  •  •  • ;  that  is,  —  c/  +  6,-  =  0,  j  =  2,  3,  •  •  • . 
Since  —  e  cos  E,  as  defined  by  (60),  is  the  integral  of  a  sine  series 
it  contains  no  constant  terms;  therefore  the  6,-  are  the  sums  of 
the  coefficients  of  the  cosine  terms.  Now  consider 


=  J[ 2'   I  1  -  c  -  e  cos  M  -  ^cos  2M  +  • 


dM. 


It  was  shown  in  Problem  4,  p.  154,  that  the  value  of  this  integral 
is  2ir(l  +  ^e2).     Therefore  the  coefficients  of  e3,  e*,  •  •  •  contain  no 

constant  terms  and  the  exact  value  of  c  is  —  4-e2. 

' 

T 

The  series  for  -  up  to  the  sixth  power  of  e  is 

"- 


100] 


DEVELOPMENT   OF  r  AND   V   IN   SERIES. 


171 


(62) 


-  =  1  -  6  cosM  -  £  (cos  2M  -  1) 
a  * 

-  2^22  (3  cos  3M  -  3  cosM) 

-  5^5  (42  cos  4M  -  4  •  22  cos  2M) 

o  !  ^5 

-  -^~  (53  cos  5M  -  5  -  33  cos  3M  +  10  cos  M) 

(64  cos  QM  -  6  •  44  cos  4Af  +  15  -  24  cos  2M) 


5!25 


The  computation  of  the  series  for  v  will  now  be  considered.     It 
follows  from  the  first  two  equations  of  (49)  that 


dv  = 


ll  -  e2 


dM, 


(l-e  cos  E)2 
which  becomes  as  a  consequence  of  Kepler's  equation 

, *(dE\rlM 

The  quantity  -=?-=.  is  found  at  once  from  (59),  and  the  result  squared 


and   integrated   gives,  after   Vl  —  e2  has  been   expanded   as   a 
power  series  in  e2, 

v  =  M  +  2e  smM  +  fe2  sin  2M 


(64) 


+  ~  (103  sin  4M  -  44  sin  2M) 
9b 

H-  jr^r  (1097  sin  5M  -  645  sin  3M  +  50  sin  M) 


jrr  (1223  sin  6M  -  902  sin  4M  +  85  sin  2M) 


When  e  is  small,  as  in  the  planetary  orbits,  these  series  are  very 
rapidly    convergent;   if   e   exceeds   0.6627  •  •  •    they   diverge,    as 


172  DIRECT  COMPUTATION   OF   POLAR   COORDINATES.  [101 

Laplace  first  showed,  for  some  values  of  M.  This  value  of  e  is 
exceeded  in  the  solar  system  only  in  the  case  of  some  of  the  comets7 
orbits,  but  developments  of  this  sort  are  not  employed  in  com- 
puting the  perturbations  of  the  comets. 

101.  Direct  Computation  of  the  Polar  Coordinates.*  It  has 
been  observed  that  there  is  considerable  labor  involved  in  finding 
the  coordinates  at  any  time  in  the  case  of  elliptic  motion.  The 
question  arises  whether  it  may  not  be  due  partly  to  the  fact  that 
the  final  result  is  obtained  by  determining  E  as  an  intermediary 
function  from  Kepler's  equation.  The  question  also  arises 
whether  the  coordinates  cannot  conveniently  be  found  directly 
from  the  differential  equations.  It  will  be  shown  that  the  answer 
to  the  latter  question  is  in  the  affirmative. 

Equations  (16)  become  in  polar  coordinates 


.<B 

On  integrating  the  second  of  these  equations  and  eliminating 

dv 

dt 


-77  from  the  first  by  means  of  the  integral,  the  result  is  found  to  be 


d?r      h* 

dt2      r3  ~*          r2 


After  eliminating  /b2(l  +  m)  by  the  first  equation  of  (48)  and 
changing  from  the  independent  variable  t  to  M  by  means  of  the 
second  equation  of  (48),  these  equations  become 


(65) 


The  first  equation  of  (65)  is  independent  of  the  second  and 
can  be  integrated  separately.  It  is  satisfied  by  r  =  a  and  e  =  0,  in 
which  case  the  orbit  is  a  circle.  In  order  to  get  the  elliptic  orbit 
let 

*This  method  was  first  published  by  the  author  in  the  Astronomical 
Journal,  vol.  25  (1907). 


101]  DIRECT  COMPUTATION   OF   POLAR   COORDINATES.  173 

(66)  r  =  a(l  -  pe), 

where  ape  is  the  deviation  from  a  circle.  When  the  planet  is  at 
perihelion,  r  =  a(l  -  e).  Therefore  p  =  1  for  M  =  0.  When  the 
planet  is  at  aphelion,  r  =  a(l  +  e).  Therefore  p  =  —  1  for 

M  =  TT,  and  p  varies  between  —  1  and  +  1.     Since  -j      is  zero 


for  M  equal  to  0  and  TT,  it  follows  that  -7^  is  zero  for  M  equal  to 

0  and  TT. 

When  (66)  is  substituted  in  (65),  these  equations  become 


= 

Since  e  is  less  than  unity  and  p  varies  from  —  1  to  +  1,  the 
second  terms  of  these  equations  can  be  expanded  as  converging 
power  series  in  e,  giving 


(67) 


It  has  been  shown  that  r,  and  hence  p,  is  expansible  as  a  power 
series  in  e.  This  fact  also  follows  from  the  form  of  the  first  equa- 
tion of  (67)  and  the  general  principles  of  Differential  Equations. 
Hence  p  can  be  written  in  the  form 

(68)  p  =  po  -f  PI  e  +  p2  e2  +  •  •  • , 

where  po,  PI,  pz,  •  •  •  are  functions  of  M  which  remain  to  be  deter- 
mined. Since  p  is  periodic  with  the  period  2w  for  all  e  less  than 
unity,  each  py  separately  is  a  sum  of  trigonometric  terms.  Since 
the  motion  is  symmetrical  with  respect  to  the  major  axis  of  the 
orbit,  and  since  M  =  0  when  the  planet  is  at  its  perihelion,  p  is 
an  even  function  of  M.  This  is  true  for  all  values  of  e  for  which 
the  series  converges,  and  therefore  each  p/  is  a  sum  of  cosine 
terms. 

A  change  in  the  sign  of  e  is  equivalent  to  changing  the  origin  to 
the  other  focus  of  the  ellipse.  Hence  if  the  sign  of  e  is  changed 
and  TT  is  added  to  M  the  value  of  r  is  unchanged;  from  (66)  it  fol- 


174 


DIRECT   COMPUTATION    OF   POLAR   COORDINATES. 


[101 


lows  that  the  sign  of  p  is  changed.  Since  this  is  true  for  all  values 
of  e  for  which  the  series  converges 

Pi(M)e*  =  -  Pi(M  +  *)(- e)t. 

Therefore  if  j  is  even  p/  is  a  sum  of  cosines  of  odd  multiples  of  M, 
and  if  j  is  odd  p/  is  a  sum  of  cosines  of  even  multiples  of  M.  It 
is  seen  on  referring  to  equations  (68)  and  (66)  that  this  is  the  same 
property  as  that  which  was  established  Art.  100. 

It  can  easily  be  proved  from  the  properties  of  the  p/  and  the 
second  equation  of  (67)  that  v  is  expressible  as  a  series  of  the  form 


(69)  v  =  VQ  +  vie  +  the2  +  •  ••, 

and  that  each  Vj  (j  >  1)  is  a  sum  of  sines  of  integral  multiples  of  M. 
A  more  detailed  discussion  shows  that  if  j  is  even  v3-  is  a  sum  of 
sines  of  even  multiples  of  M,  and  if  j  is  odd  v3-  is  a  sum  of  sines  of 
odd  multiples  of  M. 

The  solution  can  be  directly  constructed  without  any  difficulty. 
The  result  of  substituting  (68)  in  the  first  of  (67)  is 

\d2p,    j 


dM* 


d2^ 
^dM2 


+  [PO  +  Pie  +  p2e2 
>o]e  +  [3po  —  6p0pi  — 


On  equating  coefficients  of  corresponding  powers  of  e  in  the  left 
and  right  members  of  this  equation,  it  is  found  that 

-A    d*Po    i 


(70) 


(&)  dM~2+pl  =  l~  3p°2> 


(c) 


P2  =  3PO(1  -  2Pi  -  2p02), 


Equations  (70)  can  be  integrated  in  the  order  in  which  they 
are  written.     Two  constants  of  integration  arise  at  each  step 

and  they  are  to  be  determined  so  that  p  =  1  and  -~f  =  0  for 

M  =  0  whatever  may  be  the  value  of '  e.     It  follows  from  (68) 
that  these  conditions  are 

p(0)  =  po(0)  +  pi(0)e  +  p2(0)e2  +  -  - 


dp_=dpo  .dpi          dpz 
dM      dM^  dM     rdM 


101]  DIRECT   COMPUTATION    OF   POLAR   COORDINATES.  175 

where  M  is  given  the  value  0  after  the  derivatives  of  the  second 
equation  have  been  formed.  Since  these  equations  hold  for  all 
values  of  e,  it  follows  that 

fpo(O)  =  i,      PI(O)  =o,      P2(0)  =  o, 

(71)     1     dp_o  =  n  ^_i  =  n  dpz  = 

[    dM~  dM~  dM~ 

The  general  solution  of  equation  (a)  of  (70)  is  (Art.  32) 
Po  =  do  cos  M  +  60  sin  M, 

where  a0  and  bQ  are  the  constants  of  integration.  It  follows 
from  (71)  that  a0  =  1,  &o  =  C,  and  therefore  that 


Po 


=  cosM. 


The  fact  that  60  is  zero  also  follows  from  the  general  property 
that  the  pj  involve  only  cosines. 

On  substituting  the  value  of  po  in  the  right  member  of  (6)  of  (70) , 
this  equation  becomes 

+  Pi  =  —  |  —  f  cos  2M. 


dM2 

This  equation  can  be  solved  by  the  method  of  the  variation  of 
parameters  (Art.  37).  But  since  the  part  of  the  solution  which 
comes  from  the  right  member  will  contain  terms  of  the  same 
form  as  the  right  member,  it  is  simpler  to  substitute  the  expression 

Pi  =  a\  cos  M  +  61  sin  M  +  c\  +  di  cos  2M 

in  the  differential  equation  and  to  determine  Ci  and  di  so  that  it 
will  be  satisfied.  This  leads  to  the  solution 

Pi  =  «i  cos  M  +  61  sin  M  —  \  +  \  cos  2M, 

which  is  the  general  solution  since  it  satisfies  the  differential 
equation  and  has  the  two  arbitrary  constants  ai  and  61.  On 
determining  ai  and  61  by  (71),  the  expression  for  pi  becomes 

pi  =  —  i  +  |  cos  2M. 

With  the  values  of  po  and  pi  which  have  been  found  equation 
(c)  of  (70)  becomes 


of  which  the  general  solution  is 


176  DIRECT   COMPUTATION   OF   POLAR   COORDINATES.  [101 

p2  =  a2  cos  M  +  62  sin  M  +  f  cos  3M. 

If  a2  and  62  are  determined  by  (71),  the  final  expression  for  p2 
becomes 

p2  =  f  (—  cos  M  +  cos  3M). 

This  process  of  integration  can  be  continued  as  far  as  may  be 
desired.     It  follows  from  the  results  which  have  been  found  that 

^  =  1  -  pe  =  1  -  (po  +  pie  +  p2e2  +  •  •  -)e 

=  1  -  e  cos  M  -Je^cos  2M  -  1)  -  |e3(cos  3M  -  cos  M )  •  •  • , 

which  agrees  with  those  given  in  (62). 

When  the  values  for  p0,  pi,  •  •  •  are  substituted  in  the  second 
equation  of  (67),  the  result  is 

1  +  2e  cos  M  +  fe2  cos  2M  +  •  •  •, 

and  the  integral  of  this  equation  is 

v  =  c  +  M  +  2e  sin  M  +  f  e2  sin  2M  + 

Since  v  =  0  when  M  =  0,  the  arbitrary  constant  c  is  zero,  and 
the  result  agrees  with  that  given  in  (64). 

The  method  which  has  just  been  developed  is,  for  this  special 
problem,  perhaps  not  superior  to  that  depending  upon  the  solu- 
tion of  Kepler's  equation.  But  if  the  conditions  of  the  problem 
are  modified  a  little,  for  example  by  adding  the  terms  which 
would  come  from  the  oblateness  of  a  planet  when  the  body  moves 
in  the  plane  of  its  equator  [equations  (30),  Chapter  IV],  Kepler's 
equation  no  longer  holds  and  the  method  depending  on  it  fails, 
while  the  one  under  consideration  here  can  be  applied  without 
any  modification  except  in  the  numerical  values  of  the  coefficients 
which  depend  upon  the  terms  added  to  the  differential  equations. 
But  additional  terms  in  the  differential  equations  change  the 
period  of  the  motion,  if  indeed  it  remains  periodic,  and  in  order 
to  exhibit  the  periodicity  explicitly  some  modifications  of  the 
methods  of  determining  the  constants  of  integration  are  in  gen- 
eral necessary.  This  method  of  integrating  in  series  is  typical  of 
those  which  are  employed  in  the  theories  of  perturbations  and  the 
more  difficult  parts  of  Celestial  Mechanics,  and  for  this  reason 
it  should  be  thoroughly  mastered. 


102]  POSITION   IN   HYPERBOLIC   ORBITS.  177 

102.  Position  in  Hyperbolic  Orbits.  There  are  close  analogies 
between  this  problem  and  that  of  finding  the  position  of  a  body 
in  an  elliptic  orbit.  But  it  follows  from  the  polar  equation  of 
the  hyperbola, 


r  = 


1  +  €  COS  V  ' 

where  a  is  its  major  semi-axis  and  e  its  eccentricity,  that  in  this 
case  v  can  vary  only  from  —  TT  +  cos"1  (  -  J  to  -f  TT  —  cos"1  (  -  J  . 

The  integrals  of  areas  and  vis  viva  are  respectively  in  the  case 
of  hyperbolic  orbits 


l_         \     /  2  ~\  \ 

U,l 

(72)     \ 

(jt)  +r\Jt)    =A;2(1  +  m)  U+«/ 

On  eliminating  v  from  the  second  of  these  equations  by  means 
of  the  first  and  solving,  it  is  found  that 

,.  rdr 

avdt  = 

where 


This  equation  can  be  integrated  at  once  in  terms  of  hyperbolic 
functions,  but  it  is  preferable  to  introduce  first  an  auxiliary 
quantity  F  corresponding  to  the  eccentric  anomaly  in  elliptic 
orbits.  Let 

(73)  a  +  r  =  ~  (eF  +  erf)  =  ae  cosh  F; 
then 

vdt=  I  -  1  +  €-  (eF  +  e~F)}dF  =  [-  1  +  e  cosh  F]dF. 
[  z  j 

The  integral  of  this  equation  is 

(74)  M  =  v(t  -  T}  =  -  F  +  I  (eF-  e~F)  =  -  F  +  e  sinh  F, 

which  gives  t  when  F  is  known.  The  inverse  problem  of  finding  F 
when  v(t  —  T)  is  given  is  one  of  more  difficulty.  The  most 
expeditious  method  would  be,  in  general,  to  find  an  approximate 
value  of  F  by  some  graphical  process,  and  then  a  more  exact 
13 


178  POSITION   IN   ELLIPTIC   AND    HYPERBOLIC  [103 

value  by  differential  corrections.     The  value  of  F  satisfying  (74) 
is  the  abscissa  of  the  point  of  intersection  of  the  line 

y  =  i  (F  +  M), 

and  the  hyperbolic  sine  curve 

ef—  e-f 
y  =  --  -- 


The   differential   corrections   could   be   computed   in   a   manner 
analogous  to  that  developed  in  the  case  of  the  elliptic  orbits. 

From  (73)  and  the  polar  equation  of  the  hyperbola,  it  follows 
that 


r  =  =  a  _ 

1   -f-  €  COS  V 


and  from  this  equation, 


f  ~F)        /€  +  1, 

tan—  =  -v/  --  -  =\  -  -tann— 

-'       >/«  -  1 


- 
- 
2         6  -  1  V+  1  +  i(e*  +  e-')         «  -  1 


which  is  a  convenient  formula  for  computing  v  when  F  has  been 
found. 

103.  Position  in  Elliptic  and  Hyperbolic  Orbits  when  e  is  Nearly 
Equal  to  Unity.  The  analytical  solutions  heretofore  given  have 
depended  upon  expansions  in  powers  of  e.  If  e  is  large,  as  in 
the  case  of  some  of  the  periodic  comets'  orbits,  the  convergence 
ceases  or  is  so  slow  that  the  methods  become  impracticable. 
The  graphical  process,  however,  avoids  this  difficulty. 

In  order  to  obtain  a  workable  analytical  solution,  the  develop- 
ments for  elliptical  orbits  will  be  made  in  powers  of  y— - — .  The 

start  is  made  from  the  equation  of  areas  and  the  polar  equation 
of  the  orbit  which  will  be  assumed  to  be  an  ellipse. 
Let 

w  =  tan-, 
1  -  e 


then  the  equation  of  areas  becomes 


(1  +  w2) 


When  X  is  very  small  the  right  member  of  this  equation  can  be 


103]  ORBITS   WHEN   6   IS   NEARLY   EQUAL   TO   UNITY.  179 

developed  into  a  rapidly  converging  series  in  X  for  all  values  of  v 
not  too  near  180°.  Since  the  periodic  comets  are  always  invisible 
when  near  aphelion,  there  will  seldom  be  occasion  to  consider  the 
solution  in  this  region.  On  expanding  the  right  member  and 
integrating,  the  result  is  found  to  be 


2(1  - 
(75) 


When  the  orbit  is  a  parabola  e  =  I  and  X  =  0,  and  this  equation 
reduces  to  (32),  which  is  a  cubic  in  w.  Since  the  perihelion 

k 
distance  in  an  ellipse  is  q  =  a(l  —  e)  and  n  =  -j ,  it  follows  that 

n  Vl  +  e  _k  Vl  +  e 
2(1  -  e}*  ~        2q* 

It  is  desired  to  find  the  value  of  w  for  any  value  of  t.  If  the 
eccentricity  should  become  equal  to  unity,  the  left  member  keeping 
the  same  value,  equation  (75)  would  have  the  form 

(76)  fc(12+e)i  (t  _  D  =  w  +  W3, 

where  W  would  be  the  tangent  of  half  the  true  anomaly  in  the 
resulting  parabolic  orbit.  From  this  equation  W  can  be  deter- 
mined by  means  of  Barker's  tables,  or  from  equations  (33). 
Suppose  W  has  been  found;  then  w  can  be  expressed  as  a  series  in 
X  of  which  the  coefficients  are  functions  of  W.  For,  assume  the 
development 

(77)  w  =  a0  +  aiX  +  a2X2  +  a3X3  +  •  •  • ; 

substitute  it  in  the  right  member  of  (75),  which  is  equal  to  the 
right  member  of  (76).  The  result  of  the  substitution  is 

W  +  ^  =  a0  +  ^  +  [a,  +  ao2a!  -  fa03  -  |a05]X 
+  [a2  +  a02a2  +  a0ai2 

+  [a3  +  ao2a3  +  7f 

-  4a0W  +  3a04ai  +  3a06ai  -  fa07  -  |a09]  X3 


180  POSITION  IN   ELLIPTIC   AND   HYPERBOLIC   ORBITS.  [103 

Since  this  equation  is  an  identity  in  X,  the  coefficients  of  corre- 
sponding powers  of  X  are  equal.     Hence 


ai(l  +  ao2)  =  tao3  + 
a2(l  +  ao2)  =  - 


4a0W 
3a06ai  +  ^a07 


There  are  three  solutions  for  a0,  only  one  of  which  is  real.     On 
taking  the  real  root  of  the  first  equation,  it  is  found  that 


_  H^5  +  tffTF7  +  j|TF9  + 

(1  +  Tf2)3 

=    w*+ 


(1  +  TF2)5 


When  the  values  of  these  coefficients  are  substituted  in  (77)  the 
tangent  of  one-half  the  true  anomaly  is  determined.  The  first 
term  gives  that  which  would  come  from  a  parabolic  orbit,  the 
remaining  terms  vanishing  for  e  =  1.  In  the  series  (64)  the  first 
term  in  the  right  member  would  be  the  true  anomaly  if  the  orbit 
were  a  circle,  the  higher  terms  being  the  corrections  to  circular 
motion.  In  the  series  (77)  the  first  term  in  the  right  member  would 
give  the  tangent  of  one-half  the  true  anomaly  if  the  orbit  were  a 
parabola,  the  higher  terms  being  the  corrections  to  parabolic 
motion. 

These  equations  apply  equally  to  hyperbolic  orbits  in  which  the 
eccentricity  is  near  unity  if  1  —  e  and  \-\-e  are  changed  to  e  —  1 
and  e  H-  1  throughout,  where  €  is  the  eccentricity  of  the  hyperbola. 


PROBLEMS.  181 

XV.    PROBLEMS. 

1.  Show  how  the  cubic  equation  (32)  can  be  solved  approximately  for 
tan  |  with  great  rapidity  by  the  aid  of  a  graphical  construction. 

2.  Develop  the  equations  for  differential  corrections  to  the  approximate 
values  found  by  the  graphical  method.     Apply  to  a  particular  problem  and 
verify  the  result. 

3.  If  e  =  0.2  and  M  =  214°,  find  E0,  M0,  El}  M1}  E2,  and  M2. 

Ans.    E0  =  208°  39'  16".6,     M0  =  214°  8'  58".6;     Ei  =  208°  31'  38".4, 
Mi  =  213°  59'  59".8;     E2  =  208°  31'  38".6,     M2  =  214°  00'  00". 

4.  Show  from  the  curves  employed  in  solving  Kepler's  equation  that  the 
solution  is  unique  for  all  values  of  e  <  1  and  M. 

5.  In  (50)  the  quadrant  is  not  determined  by  the  equation;  show  that 
corresponding  values  of  \v  and  \E  always  lie  in  the  same  quadrant. 

6.  Express  the  rectangular  coordinates  x  =  r  cos  v,  y  —  r  sin  v  in  terms 
of  the  eccentric  anomaly,  and  then,  by  means  of  the  Lagrange  expansion 
formula,  in  terms  of  M. 

2  =  cos  M  +  I  (cos  2M  -  3)  +  £^  (3  cos  3Af  -  3  cos  M) 


^7^  (42  cos  4M  -  4  •  22  cos  2Af )  + 
Ans. 

"'  =  sin  M  +  ^sin  2M  +  ^  (32  sin  3M  -  15  sin  M) 

(43  sin  4M  -  10  •  23  sin  2M)  + 

7.  Show  that  the  properties  of  E  as  a  power   series  in  e,  which  were 
established  in  Art.  99,  follow  from  the  Lagrange  expansion. 

8.  Derive  the  first  three  terms  of  the  series  for  r  by  the  Lagrange  formula. 

9.  Give  a  geometrical  interpretation  of  F  (Art.  102)  corresponding  to  that 
of  E  in  an  elliptic  orbit. 

10.  Express  v  as  a  power  series  in  e  by  a  method  analogous  to  that  used  in 
Art.  103. 

11.  Show  that  the  branch  of  the  hyperbola  which  is  convex  to  the  sun  is 
described  by  the  body  in  purely  imaginary  time. 

12.  Add  to  the  right  members  of  equations  (16)  the  terms  —  TQ  (1  +w)62ei2^ 

o 

and  —  —  (1  +  m)62ei2  ^  ,  which  come  from  the  oblateness  of  the  central  body 

[equations  (30),  Chap,  iv.],  where  e\  is  the  eccentricity  of  a  meridian  section, 
and  integrate  by  the  method  of  Art.  101. 


182  THE   HELIOCENTRIC   POSITION  [104 

104.  The  Heliocentric  Position  in  the  Ecliptic  System.  Methods 
have  been  given  for  finding  the  positions  in  the  orbits  in  the 
various  cases  which  arise.  The  formulas  will  now  be  derived 
for  determining  the  position  referred  to  different  systems  of  axes. 
The  origin  will  first  be  kept  fixed  at  the  body  with  respect  to 
which  the  motion  of  the  second  is  given.  Since  most  of  the  appli- 
cations are  in  the  solar  system  where  the  origin  is  at  the  center  of 
the  sun,  the  coordinates  will  be  called  heliocentric. 

Positions  of  bodies  in  the  solar  system  are  usually  referred  to 
one  of  two  systems  of  coordinates,  the  ecliptic  system,  or  the 
equatorial  system.  The  fundamental  plane  in  the  ecliptic  system 
is  the  plane  of  the  earth's  orbit;  in  the  equatorial  system  it  is  the 
plane  of  the  earth's  equator.  The  zero  point  of  the  fundamental 
circles  in  both  systems  is  the  vernal  equinox,  or  the  point  at  which 
the  ecliptic  cuts  the  equator  from  south  to  north,  and  is  denoted 
by  V.  The  polar  coordinates  in  the  ecliptic  system  are  called 
longitude  and  latitude;  and  in  the  equatorial,  right  ascension  and 
declination.  When  the  origin  is  at  the  sun  Roman  letters  are 
used  to  represent  the  coordinates,  and  when  at  the  earth,  Greek. 
Thus 

Origin  at  sun.  Origin  at  earth. 

longitude  I  .      X  measured  eastward, 

latitude  b  0  +  if  north;  —  if  south, 

right  ascension  a  a  measured  eastward, 

declination  d  d  +  if  north;  —  if  south, 

distance  r  p 

In  practice  a  and  d  are  very  seldom  used.  Absolute  positions  of 
fundamental  stars  are  given  in  the  equatorial  system,  and  the 
observed  positions  of  comets  are  determined  by  comparison  with 
them.  In  some  theories  relating  to  planets  and  comets,  especially 
in  considering  the  mutual  perturbation  of  planets  and  their  per- 
turbations of  comets,  it  is  more  convenient  to  use  the  ecliptic 
system;  hence  it  is  necessary  to  be  able  to  transform  the  equations 
from  one  system  to  the  other. 

The  ascending  node  is  the  projection  on  the  ecliptic,  from  the 
sun,  of  the  place  at  which  the  body  crosses  the  plane  of  the  ecliptic 
from  south  to  north.  It  is  measured  from  a  fixed  point  in  the 
ecliptic,  the  vernal  equinox,  and  is  denoted  by  <&.  The  projection 
of  the  point  where  the  body  crosses  the  plane  of  the  ecliptic  from 
north  to  south  is  called  the  descending  node,  and  is  denoted  by  t3> 


104]  IN   THE  ECLIPTIC   SYSTEM.  183 

The  inclination  is  the  angle  between  the  plane  of  the  orbit  and 
the  plane  of  the  ecliptic,  and  is  denoted  by  i.  It  has  been  the 
custom  of  some  writers  to  take  the  inclination  always  less  than 
90°,  and  to  define  the  direction  of  motion  as  direct  or  retrograde, 
according  as  it  is  the  same  as  that  of  the  earth  or  the  opposite. 
Another  method  that  has  been  used  is  to  consider  all  motion  direct 
and  the  inclination  as  varying  from  0°  to  180°.  The  latter  method 
avoids  the  use  of  double  signs  in  the  formulas  and  is  adopted  here. 
[See  Art.  86.]  The  node  and  inclination  define  the  position  of 
the  plane  of  the  orbit  in  space. 

The  distance  from  the  ascending  node  to  the  perihelion  point 
counted  in  the  direction  of  the  motion  of  the  body  in  its  orbit  is  w, 
and  defines  the  orientation  of  the  orbit  in  its  plane.  The  longitude 
of  the  perihelion  is  denoted  by  TT,  and  is  given  by  the  equation 


This  element  is  not  a  longitude  in  the  ordinary  sense  because  it 
is  counted  in  two  different  planes. 

The  problem  of  relative  motion  of  two  bodies  was  of  the  sixth 
order  (Art.  85)  ,  and  in  the  integration  six  arbitrary  constants  were 
introduced.  There  are  six  elements,  therefore,  which  are  inde- 
pendent functions  of  these  constants.  They  are 

a  =  major  semi-axis,  which  defines  the  size  of  the  orbit  and 

the  period  of  revolution. 

e  =  the  eccentricity,  which  defines  the  shape  of  the  orbit. 
&>  =  longitude  of  ascending  node,  and 
i  =  inclination  to  plane  of  the  ecliptic,  which  together  define 

the  position  of  the  plane  of  the  orbit. 

a)  =  longitude  of  the  perihelion  point  measured  from  the  node, 
or  TT  =  longitude  of  the  perihelion,  either  defining  the 
orientation  of  the  orbit  in  its  plane. 

T  =  time  of  perihelion  passage,  defining,  with  the  other  ele- 

ments, the  position  of  the  body  in  its  orbit  at  any  time. 

The  polar  coordinates  have  been  computed;  hence  the  rect- 

angular coordinates  with  the  positive  end  of  the  re-axis  directed  to 

the  perihelion  point  and  the  i/-axis  in  the  plane  of  the  orbit  are 

given  by  the  equations 


(78) 


184 


HELIOCENTRIC   POSITION   IN    ECLIPTIC   SYSTEM. 


[104 


If  the  x-axis  is  rotated  backward  to  the  line  of  nodes,  the  coordinates 
in  the  new  system  are 

fx  =  r  cos  (v  +  co)  =  r  cos  (v  +  TT  —  ft), 
y  =  r  sin  (v  +  o>)  =  r  sin  (t;  +  TT  —  ft), 
2=0. 

The  longitude  of  the  body  in  its  orbit  counted  from  the  ascending 
node  is  called  the  argument  of  the  latitude  and  is  denoted  by  u. 
It  is  given  by  the  equation 

u  =  v  -\-  w, 
hence  u  is  known  when  v  has  been  found. 


Fig.  30. 


Let  S  represent  the  sun  and  Sxy  the  plane  of  the  ecliptic;  S&A, 
the  plane  of  the  orbit;  ft,  the  ascending  node;  n,  the  perihelion 
point;  A,  the  projection  of  the  position  of  the  body;  and  angle 
USA  =  v.  Then  ftA  =  co  +  v  =  u. 

Let  the  position  of  the  body  now  be  referred  to  a  rectangular 
system  of  axes  with  the  origin  at  the  sun,  the  x-axis  in  the  line  of 
the  nodes,  and  the  i/-axis  in  the  plane  of  the  ecliptic.  Then  equa- 
tions (79)  become 

x'  =  r  cos  (v  +  w)  =  r  cos  u, 
(80)          •{  y'  =  r  sin  (v  +  «)  cos  i  =  r  sin  u  cos  i, 
z'  =  r  sin  (v  +  o>)  sin  i  =  r  sin  u  sin  i. 


105]  TRANSFER   OF   ORIGIN    TO    THE   EARTH.  185 

But,  in  terms  of  the  heliocentric  latitude  and  longitude, 
-x'  =  r  cos  b  cos  (I  —  &), 

(81)  -  y'  =  r  cos  b  sin  (I  -&), 

.  z'  —  r  sin  b. 

Therefore,  comparing  (80)  and  (81),  it  is  found  that 
r  cos  b  cos  (I  —  (fi> )  =  cos  u, 

(82)  -j  cos  b  sin  (I  —  &)  =  sin  u  cos  t, 

I  sin  b  =  sin  it  sin  i\ 

whence 

ftan  (Z  —  &)  =  tan  it  cos  i, 

(83)  -i 

[  tan  6  =  tan  i  sin  (I  —  &). 

Since  cos  6  is  always  positive,  equations  (82)  and  (83)  determine 
the  heliocentric  longitude  and  latitude,  I  and  6,  uniquely  when 
&>,  i,  and  u  are  known. 

105.  Transfer  of  the  Origin  to  the  Earth.  Let  E,  H,  Z  be  the 
geocentric  coordinates  of  the  center  of  the  sun  referred  to  a  system 
of  axes  with  the  x-axis  directed  to  the  vernal  equinox,  and  the 
?/-axis  in  the  plane  of  the  ecliptic.  Let  P,  A,  and  B*  represent  the 
geocentric  distance,  longitude,  and  latitude  of  tl*8  sun  respectively. 
These  quantities  are  given  in  the  Nautical  Almanac  for  every  day 
in  the  year.  The  rectangular  coordinates  are  expressed  in  terms 
of  them  by 

f*A  =  P  cos  B  cos  A, 
H  =  P  cos  B  sin  A, 
Z  =  P  sin  B. 

The  angle  B  is  generally  less  than  a  second  of  arc,  and  unless  great 
accuracy  is  required  these  equations  may  be  replaced  by 

H  =  P  cos  A, 
H  =  P  sin  A, 
Z  =  0. 

Let  £",  77",  and  r"  be  the  geocentric,  and  x",  y",  and  z"  the 
heliocentric,  coordinates  of  the  body  with  the  o^axis  directed 
toward  the  vernal  equinox  and  the  i/-axis  in  the  plane  of  the  eclip- 
tic. Therefore 

*  P,  A,  B  =  capital  p,  X,  ft. 


186  TRANSFORMATION   TO    GEOCENTRIC    COORDINATES.  [106 

«"  -  x"  +  H, 
n"  -  y"  +  H, 

r"  =  z"  +  z. 

In  polar  coordinates  these  equations  are 

•p  cos  |8  cos  X  =  r  cos  6  cos  I  +  P  cos  B  cos  A, 
p  cos  jS  sin  X  =  r  cos  6  sin  I  +  P  cos  B  sin  A, 
p  sin  j8  =  r  sin  6  +  P  sin  B. 

From  these  equations  X  and  ft  can  be  found;  but  this  system  may 
be  transformed  into  one  which  is  more  convenient  by  multiplying 
the  first  equation  by  cos  A,  the  second  by  sin  A,  and  adding  the 
products;  and  then  multiplying  the  first  by  —  sin  A  and  the 
second  by  cos  A,  and  adding  the  products.  The  results  are 

fp  cos  ft  cos  (X  —  A)  =  r  cos  b  cos  (I  —  A)  -f-  P  cos  B, 
p  cos  ft  sin  (X  —  A)  =  r  cos  b  sin  (I  —  A), 
P  sin  ft  =  r  sin  b  +  P  sin  B. 

These  equations  give  the  geocentric  distance,  longitude,  and 
latitude,  p,  X,  and  ft. 

106.  Transformation    to    Geocentric    Equatorial    Coordinates. 

Let  e  represent  the  inclination  of  the  plane  of  the  ecliptic  to  the 
plane  of  the  equator.  Let  £",  rj",  and  f"  be  the  geocentric  co- 
ordinates of  the  body  referred  to  the  ecliptic  system  with  the 
x-axis  directed  toward  the  vernal  equinox.  Then,  the  equatorial 
system  can  be  obtained  by  rotating  the  ecliptic  system  around  the 
x-axis  in  the  negative  direction  through  the  angle  e,  the  relations 
between  the  coordinates  in  the  two  systems  being 

-r, 

]'"  =  77"  cos  €  -  f"  sin  c, 
'"  =  i?"sine  +  r"  cose; 
or,  in  polar  coordinates, 

•  cos  8  cos  a  =  cos  ft  cos  X, 

(86)       -  cos  6  sin  a  =  cos  ft  sin  X  cos  e  —  sin  ft  sin  e, 
.  sin  5  =  cos  ft  sin  X  sin  e  +  sin  ft  cos  e. 

In  order  to  solve  these  equations  conveniently  for  5  and  a  the 
auxiliaries  n  and  N  will  be  introduced  by  the  equations 


107] 
(87) 


COMPUTATION    OF   GEOCENTRIC   COORDINATES. 


187 


n  sin  AT  =  sin  0, 
n  cos  N  =  cos  j8  sin  X, 
in  which  n  is  a  positive  quantity.     Then  equations  (86)  become 
'  cos  8  cos  a  =  cos  |8  cos  X, 
cos  5  sin  a  =  n  cos  (N  -f-  e) , 
^sin  6  =  n  sin  (JV  +  e) ; 

n  sin  A7"  =  sin  0, 

n  cos  TV  =  cos  j8  sin  X, 

cos  (TV  +  e)  tan  X 
tan  a.  =  — 


whence 


(88) 


cos  N 
tan  6  =  tan  (N  +  e)  sin  a. 

These  equations,  together  with  the  first  of  (86),  which  is  used  in 
determining  the  quadrant  in  which  a  lies,  give  a  and  8  without 
ambiguity  when  X  and  0  are  known. 

If  a  and  8  are  given  and  X  and  /3  are  required,  the  equations  from 
which  they  can  be  computed  are  found  by  interchanging  a  and  5 
with  X  and  /?,  and  changing  e  to  —  e  in  (88).  They  are* 

m  sin  M  =  sin  8, 

m  cos  M  =  cos  8  sin  a, 

cos  (M  —  e)  tan  a 
tan  X  =  -  ,:.        - , 

cos  M 

tan  j8  =  tan  (M  —  e)  sin  X. 

107.  Direct  Computation  of  the  Geocentric  Equatorial  Co- 
ordinates. The  geocentric  equatorial  coordinates,  a  and  8,  can 
be  found  directly  from  the  elements,  i  and  &,  and  the  argument 
of  the  latitude  u}  without  first  finding  the  ecliptic  coordinates, 
X  and  |S. 

In  a  system  of  axes  with  the  z-axis  directed  to  the  node  and  the 
7/-axis  in  the  plane  of  the  ecliptic,  the  equations  for  the  heliocentric 
coordinates  are 

x'  =  r  cos  u, 

y'  =  r  sin  u  cos  i, 


z    =  r  sin  u  sin  i. 


*  m  and  M  are  new  auxiliaries,  not  being  related  to  any  of  the  quantities 
which  these  letters  previously  have  represented. 


188 


COMPUTATION  OF  GEOCENTRIC  COORDINATES. 


[107 


If  the  system  is  rotated  around  the  z-axis  until  the  z-axis  is  directed 
toward  the  vernal  equinox,  the  coordinates  are 

'x"  =  x'  cos  ft  —  y'  sin  ft, 
y"  =  x'  sin  ft  +  y'  cos  ft, 


or, 


(90) 


= 


' x"  =  r  (cos  u  cos  ft  —  sin  i£  cos  i  sin  ft), 
"  =  r  (cos  w  sin  ft  -f-  sin  u  cos  i  cos  ft), 


r  sin  w  sin  i. 


If  the  system  is  rotated  now  around  the  z-axis  through  the  angle 
—  e,  the  coordinates  become 


==  y    cos  e  —  z    sin  e, 
X"  =  y"  sin  €  +  z"  cos  e; 
or,  in  polar  coordinates, 

;'"  =  rjcos  u  cos  ft  —  sin  u  cos  i  sin  ft }, 
'"'  =  r{  (cos  w  sm  ft  4~  sin  it  cos  i  cos  ft)  cos  e 
(91)     •{  —  sin  w  sin  i  sin  e}, 

z/"  =  r{(cos  u  sin  ft  +  sin  u  cos  i  cos  ft)  sin  e 

+  sin  u  sin  i  cos  e } . 

In  order  to  facilitate  the  computation  Gauss  introduced  the  new 
auxiliaries  A,  a,  B,  6,  C,  and  c  by  the  equations 

sin  a  sin  A  =  cos  ft , 

sin  a  cos  A  =  —  sin  ft  cos  it  sin  a  >  0, 

sin  b  sin  B  =  sin  ft  cos  e,  sin  6  >  0, 

sin  b  cos  B  =  cos  ft  cos  i  cos  e  —  sin  i  sin  e, 
sin  c  sin  C  =  sin  ft  sin  e,  sin  c  >  0, 

L  sin  c  cos  C  =  cos  ft  cos  i  sin  e  +  sin  i  cos  c. 

These  constants  depend  upon  the  elements  alone,  so  they  need  be 
computed  but  once  for  a  given  orbit.  They  are  of  particular 
advantage  when  the  coordinates  are  to  be  computed  for  a  large 
number  of  epochs,  as  in  constructing  an  ephemeris.  When  these 


(92) 


PROBLEMS.  189 

constants  are  substituted  in  (91),  these  equations  for  the  helio- 
centric coordinates  take  the  simple  form 

x'"  —  r  sin  a  sin  (A+  u), 


(93) 


y'"  =  r  sin  b  sin  (B  +  u), 
z'"  =  r  sin  c  sin  (C  +  w), 


from  which  x"',  ?/'",  and  2'"  can  be  found. 

Then  finally,  the  geocentric  equatorial  coordinates  are  defined 
by 

p  cos  5  cos  a  =  x'"  +  X', 
(94)  -  p  cos  5  sin  a  =  y'"  +  Y7, 


where  X',  Y',  and  Z'  are  the  rectangular  geocentric  coordinates  of 
the  sun  referred  to  the  equatorial  system.  They  are  given  in  the 
Nautical  Almanac  for  every  day  in  the  year,  and,  therefore,  these 
equations  define  p,  a,  and  5. 

This  completes  the  theory  of  the  determination  of  the  helio- 
centric and  geocentric  coordinates  of  a  body,  moving  in  any  orbit, 
when  either  the  ecliptic  or  the  equatorial  system  is  used. 


XVI.     PROBLEMS. 

1.  Interpret  the  angle  N,  equation  (87),  geometrically  and  show  that  n  is 
simply  a  factor  of  proportionality. 

2.  Suppose  the  ascending  node  is  taken  always  as  that  one  which  is  less 
than  180°,  and  that  the  inclination  varies  from  —  90°  to  +  90°;  discuss  the 
changes  which  will  be  made  in  the  equations  (78),  •  •  •,  (93),  and  in  particular 

write  the  definitions  of  the  Gaussian  constants  a,  A, ,  C  f or  this  method 

of  defining  the  elements. 

3.  Interpret  the  Gaussian  constants,  defined  by  (92),  geometrically. 


190  HISTORICAL   SKETCH. 


HISTORICAL  SKETCH  AND  BIBLIOGRAPHY. 

The  Problem  of  Two  Bodies  for  spheres  of  finite  size  was  first  solved  by 
Newton  about  1685,  and  is  given  in  the  Principia,  Book  i.,  Section  11.  The 
demonstration  is  geometrical.  The  methods  of  the  Calculus  were  cultivated 
with  ardor  in  continental  Europe  at  the  beginning  of  the  18th  century,  but 
Newton's  system  of  Mechanics  did  not  find  immediate  acceptance;  indeed, 
the  French  clung  to  the  vortex  theory  of  Descartes  (1596-1650)  until  Vol- 
taire, after  his  visit  to  London  1727,  vigorously  supported  the  Newtonian 
theory,  1728-1738.  This,  with  the  fact  that  the  English  continued  to 
employ  the  geometrical  methods  of  the  Principia,  delayed  the  analytical 
solution  of  the  problem.  It  was  probably  accomplished  by  Daniel  Bernouilli 
in  the  memoir  for  which  he  received  the  prize  from  the  French  Academy  in 
1734,  and  it  was  certainly  solved  in  detail  by  Euler  in  1744  in  his  Theoria 
motuum  planetarum  et  cometarum.  Since  that  time  the  modifications  have 
been  chiefly  in  the  choice  of  variables  in  which  the  problem  has  been  expressed. 

The  solution  of  Kepler's  equation  naturally  was  first  made  by  Kepler 
himself.  The  next  was  by  Newton  in  the  Principia.  From  a  graphical 
construction  involving  the  cycloid  he  was  able  to  find  very  easily  the  approxi- 
mate solution  for  the  eccentric  anomaly.  A  very  large  number  of  analytical 
and  graphical  solutions  have  been  discovered,  nearly  every  prominent  mathe- 
matician from  Newton  until  the  middle  of  the  last  century  having  given  the 
subject  more  or  less  attention.  A  bibliography  containing  references  to  123 
papers  on  Kepler's  equation  is  given  in  the  Bulletin  Astronomique,  Jan.  1900, 
and  even  this  extended  list  is  incomplete. 

The  transformations  of  coordinates  involve  merely  the  solutions  of  spherical 
triangles,  the  treatment  of  which  in  a  perfectly  general  form  the  mathematical 
world  owes  to  Gauss  (1777-1855),  and  which  was  introduced  into  American 
Trigonometries  by  Chauvenet. 

The  Problem  of  Two  Bodies  is  treated  in  every  work  on  Analytical  Me- 
chanics. The  reader  will  do  well  to  consult  further  Tisserand's  Mec.  CeL, 
vol.  i.,  chapters  vi.  and  VH. 


CHAPTER  VI. 

THE  DETERMINATION   OF   ORBITS. 

108.  General  Consideration.  In  discussing  the  problem  of 
two  bodies  [Arts.  86-88]  it  was  shown  how  the  constants  of  inte- 
gration which  arise  when  the  differential  equations  are  solved  can 
be  determined  in  terms  of  the  original  values  of  the  coordinates 
and  of  the  components  of  velocity;  and  then  it  was  shown  how 
the  elements  of  the  conic  section  orbit  can  be  determined  in  terms 
of  these  constants.  Consequently,  it  is  natural  to  seek  to  deter- 
mine the  position  and  components  of  the  observed  body  at  some 
epoch.  The  difficulty  arises  from  the  fact  that  the  observations, 
which  are  made  from  the  moving  earth,  give  only  the  direction  of 
the  object  as  seen  by  the  observer,  and  furnish  no  direct  informa- 
tion respecting  its  distance.  An  observation  of  apparent  position 
simply  determines  the  fact  that  the  body  is  somewhere  on  one 
half  of  a  defined  line  passing  through  the  observer.  The  position 
of  the  body  in  space  is  therefore  not  given,  and,  of  course,  its 
components  of  velocity  are  not  determined.  It  becomes  necessary 
on  this  account  to  secure  additional  observations  at  other  times. 
In  the  interval  of  time  before  the  second  observation  is  made  the 
earth  will  have  moved  and  the  observed  body  will  have  gone  to 
another  place  in  its  orbit.  The  second  observation  simply  deter- 
mines another  line  on  which  the  body  is  located  at  another  date. 
It  is  clear  that  the  problem  of  finding  the  position  of  the  body  and 
the  elements  of  its  orbit  from  such  data  presents  some  difficulties. 

The  first  question  to  settle  is  naturally  the  number  of  obser- 
vations which  are  necessary  in  order  that  it  shall  be  possible  to 
determine  the  elements  of  the  orbit.  Since  an  orbit  is  defined  by 
six  elements,  it  follows  that  six  independent  quantities  must  be 
given  by  the  observations  in  order  that  the  elements  may  be  de- 
termined. A  single  complete  observation  gives  two  quantities,  the 
angular  coordinates  of  the  body.  Therefore  three  complete  obser- 
vations are  just  sufficient,  so  far  as  these  considerations  are  con- 
cerned, to  define  its  orbit.  It  is  at  least  certain  that  no  smaller 
number  will  suffice.  If  the  observed  body  is  a  comet  whose 

191 


192  INTERMEDIATE   ELEMENTS.  [109 

orbit  is  a  parabola,  the  eccentricity  is  unity  and  only  five  elements 
are  to  be  found.  In  this  case  two  complete  observations  and  one 
observation  giving  one  of  the  two  angular  coordinates  are  enough. 

109.  Intermediate  Elements.  The  apparent  positions  of  the 
observed  body  are  usually  obtained  by  measuring  its  angular 
distances  and  directions  from  neighboring  fixed  stars.  Since  the 
stars  are  catalogued  in  right  ascension  and  declination  the  results 
come  out  in  these  coordinates,  but  they  can,  of  course,  be  changed 
to  the  ecliptic  system,  or  any  other,  if  it  is  desired. 

Suppose  the  observations  are  made  at  the  times  ti,  t2,  and  £3, 
and  let  the  corresponding  coordinates  be  denoted  by  their  usual 
symbols  having  the  subscripts  1,  2,  and  3  respectively.  The  right 
ascensions  and  declinations  are  functions  of  the  elements  of  the 
orbit  and  the  dates  of  observation.  These  relations  may  be 
represented  by 

'on  =  <?(&,  i,  co,  a,  6,  T]  ti), 

«2  =  <?(&,  i,  co,  a,  e,  T;  «2), 
,  i,  co,  a,  e,  T;  t8), 


,  i,  co,  a,  6,  T;  J2), 
<53  =  <K&,  i,  co,  a,  e,  T;  U). 


The  problem  consists  in  solving  these  six  equations  for  the  six 
unknown  elements.  The  functions  $  and  \f/  are  highly  transcen- 
dental and  involve  the  elements  in  a  very  complicated  fashion. 
In  the  case  of  an  ellipse  the  position  in  the  orbit  is  found  by  passing 
through  Kepler's  equation,  in  the  hyperbola  the  process  is  similar, 
and  in  the  parabola  a  cubic  equation  must  be  solved;  and  in  all 
three  cases  the  coordinates  with  respect  to  the  earth  are  obtained 
by  a  number  of  trigonometrical  transformations.  Hence  it  is 
clear  that  there  is  no  direct  solution  of  equations  (1)  by  ordinary 
processes. 

Although  the  ultimate  object  is  to  determine  the  elements  of 
the  orbit,  the  problem  of  finding  other  quantities  which  define  the 
elements  may  be  treated  first.  These  quantities  may  be  con- 
sidered as  being  intermediate  elements.  It  has  been  remarked 
that  if  the  coordinates  and  the  components  of  velocity  are  known 
at  any  epoch,  the  elements  can  be  found.  Suppose  it  is  desired 
to  find  the  polar  coordinates  and  their  derivatives,  which  deter- 


109] 


INTERMEDIATE    ELEMENTS. 


193 


mine  uniquely  the  rectangular  coordinates  and  their  derivatives, 
at  the  time  of  the  second  observation  tz.  The  equations  corre- 
sponding to  (1)  become  for  this  problem 

i  =  /  (az,  8z,  pz,  oiz ',  82',  pz'',  t\,  tz), 


(2) 


where 


as  =  f  (az,  8z,  P2,  OLZ,  8z',  pz',  tz,  ts), 

81  =  g(az,  8z,  P2,  Wi  52',  p/;  ti,  tz), 
62  =  82, 

83    =   0(«2,  52,  P2,  «2',  V,  P2';  *2,   £3), 


,_da 


dS  ,      dp 

P2'  =  _     at    t  =  U. 


Since  «2  and  62  are  observed  quantities  only  the  first,  third,  fourth, 
and  sixth  equations  are  to  be  solved  for  the  four  unknowns  P2,  ctz, 
62',  and  P27.  The  problem  is  therefore  reduced  to  the  solution  of 
four  simultaneous  equations,  and  they  are  moreover  much  simpler 
than  (1).  These  equations  can  be  put  in  a  manageable  form,  and 
this  is,  in  fact,  one  of  the  methods  of  treating  the  problem.  It  was 
first  developed  and  applied  to  the  actual  determination  of  orbits 
by  Laplace  in  1780,  and  it  has  been  somewhat  extended  and 
modified  as  to  details  by  many  later  writers. 

As  another  set  of  intermediate  elements  the  three  coordinates  at 
two  epochs  may  be  taken.  Suppose  the  times  t\  and  £3  are  chosen 
for  this  purpose.  Then  the  fundamental  equations  corresponding 
to  (1)  can  be  written  in  the  form 


(3) 


F(ai,  di,  pi,  0:3,  53, 


62  =  G(ai, 
8s  =  8s. 


pi,  0:3,  8S, 


In  this  case  the  equations  are  reduced  to  two  in  the  two  unknowns 
Pi  and  p3,  and  they  also  can  be  solved.  This  is  the  line  of  attack 
on  the  problem  laid  out  by  Lagrange  in  1778,  taken  up  inde- 
pendently and  carried  out  differently  by  Gauss  in  1801,  and  fol- 
lowed more  or  less  closely  by  many  later  writers.  In  spite  of  the 
14 


.194  PREPARATION   OF   THE   OBSERVATIONS.  [110 

hundreds  of  papers  which  have  been  written  on  the  theory  of  the 
determination  of  orbits,  very  little  that  is  really  new  or  theoreti- 
cally important  has  been  added  to  the  work  of  Laplace  and  Gauss 
unless  more  than  three  observations  are  used. 

110.  Preparation  of  the  Observations.  Whatever  method  it 
may  be  proposed  to  follow,  the  observations  as  obtained  by  the 
practical  astronomer  require  certain  slight  corrections  which  should 
be  made  before  the  computation  of  the  orbit  is  undertaken. 

The  attractions  of  the  moon  and  the  sun  upon  the  equatorial 
bulge  of  the  earth  cause  a  small  periodic  oscillation  and  a  slow 
secular  change  in  the  position  of  the  plane  of  its  equator.  Since 
the  equinoxes  are  the  places  where  the  equator  and  ecliptic  inter- 
sect, the  vernal  equinox  undergoes  small  periodic  oscillations 
(the  nutation)  and  slowly  changes  its  position  along  the  ecliptic 
(the  precession).  It  is  obviously  necessary  to  have  all  the  obser- 
vations referred  to  the  same  coordinate  system,  and  it  is  customary 
to  use  the  mean  equinox  and  position  of  the  equator  at  the  begin- 
ning of  the  year  in  which  the  observations  are  made. 

The  observed  places  are  also  affected  by  the  aberration  of  light 
due  to  the  revolution  of  the  earth  around  the  sun  and  to  its  rota- 
tion on  its  axis.  Since  the  rotation  is  very  slow  compared  to  the 
revolution,  the  aberration  due  to  the  former  is  relatively  small 
and  generally  may  be  neglected,  especially  if  the  observations 
are  not  very  precise. 

Suppose  do  and  50  are  the  observed  right  ascension  and  declina- 
tion of  the  body  at  any  time.  Then  the  right  ascension  and 
declination  referred  to  the  mean  equinox  of  the  beginning  of  the 
year,  and  corrected  for  the  annual  aberration,  are 

(a  =  ao  —  15/  —  g  sin  (G-\-a0)  tan  BQ  —  h  sin  (H +0:0)  sec  do, 
d  =  50  —  i  cos  50  —  g  cos  (G  +  «o)  —  h  cos  (H  +  a0)  sin  50, 

where  /,  g,  h,  G,  and  H  are  auxiliary  quantities,  called  the  Inde- 
pendent Star-Numbers,  which  are  given  in  the  American  Ephem- 
eris  and  Nautical  Almanac  for  every  day  of  the  year.  In 
practice  these  numbers  are  to  be  taken  from  the  Ephemeris. 
They  depend  upon  the  motions  of  the  earth,  but  their  derivation 
belongs  to  the  domain  of  Spherical  and  Practical  Astronomy, 
and  cannot  be  taken  up  here.*  The  corrections  to  ao  and  60 
furnished  by  equations  (4)  are  expressed  in  seconds  of  arc. 

*  Chauvenet,  Spherical  and  Practical  Astronomy,  vol.  i.,  chap.  xi. 


Ill]  OUTLINE    OF   THE   LAPLACIAN   METHOD.  195 

The  corrections  for  the  diurnal  aberration  are 

a  =  -  0".322  cos  <p  cos  (6  -  a0)  sec  50, 


I  A5  =  —  0".322  cos  <p  sin  (B  —  a0)  sin  50, 

where  <p  is  the  latitude  of  the  observer,  and  B  —  0:0  is  the  hour 
angle  of  the  object  at  the  time  of  the  observation.  The  second 
of  these  corrections  cannot  exceed  the  small  quantity  0".322, 
and  the  first  is  also  small  unless  50  is  near  =*=  90°. 

111.  Outline  of  the  Laplacian  Method  of  Determining  an  Orbit. 
Before  entering  on  the  details  which  are  necessary  for  the  deter- 
mination of  the  elements  of  an  orbit  by  either  of  the  two  methods 
which  are  in  common  use,  a  brief  exposition  of  the  general  lines  of 
argument  used  in  them  will  be  given.  From  these  outlines  the 
plan  of  attack  can  be  understood,  and  then  the  bearings  of  the 
detailed  investigations  will  be  fully  appreciated. 

In  order  to  keep  to  the  central  thought  suppose  only  three  com- 
plete observations  are  available  for  the  determination  of  the  orbit. 
Let  the  dates  of  the  observations  be  ti,  fa,  and  ts,  and  hence  at 
these  times  the  right  ascensions  and  declinations  of  the  observed 
body  as  seen  from  the  earth  are  known.  For  the  sake  of  definite- 
ness  in  the  terminology  let  C  represent  the  observed  body  revolv- 
ing around  the  sun,  $,  and  observed  from  the  earth  E]  £,  77,  £  the 
rectangular  coordinates  of  C  with  respect  to  E;  x,  y,  z  the  rectan- 
gular coordinates  of  C  with  respect  to  S;  X,  Y,  Z  the  rectangular 
coordinates  of  S  with  respect  to  E;  p  the  distance  from  E  to  C; 
r  the  distance  from  S  to  C;  R  the  distance  from  E  to  S.  Then 

i£  =  p  cos  5  cos  a  =  p  X, 
r]  =  p  cos  5  sin  a  =  PM, 
£  =  p  sin  6  =  p  v. 

The  quantities  X,  ju,  and  v,  which  are  the  direction  cosines  of  the 
line  from  E  to  C,  are  known  at  t\,  fa,  and  fa.  The  distance  p  is 
entirely  unknown. 

First  Step.  The  first  step  is  to  determine  the  values  of  the 
first  and  second  derivatives  of  X,  /*,  v,  X,  Y,  and  Z  at  some  time 
near  the  dates  of  observation,  say  at  fa.  It  will  be  sufficient  at 
present  to  show  that  it  can  be  done  with  considerable  approxi- 
mation without  discussing  the  best  method  of  doing  it.  The 
value  of  the  first  derivative  of  X  during  the  interval  fa  to  fa  averages 

x'  X2   —  Xi 

Al2   -  ~ , 


196  OUTLINE    OF   THE   LAPLACIAN   METHOD.  [Ill 

and  this  is  very  nearly  the  value  of  X'  at  the  middle  of  the  interval 
unless  X'  happens  to  be  changing  very  rapidly.  The  approxima- 
tion is  better  the  shorter  the  interval.  In  a  similar  manner  X^ 
is  formed.  When  the  interval  t->  —  ti  equals  the  interval  £3  —  t% 
the  value  of  X'  at  tz  is  very  nearly 

Xz    ~    2lXi2  ~T  X23J. 

If  the  intervals  are  not  equal,  adjustment  for  the  disparity  can  of 
course  be  made. 

In  a  similar  manner  it  follows  from  the  definition  of  a  derivative 
that  the  second  derivative  of  X  at  £2,  in  case  the  two  intervals  are 
equal,  is  approximately 


The  first  and  second  derivatives  of  M  and  v  are  given  approximately 
by  similar  formulas,  and  it  is  to  be  understood  that  when  the 
intervals  are  as  short  as  they  generally  are  in  practice  the  approxi- 
mations, especially  as  obtained  by  the  more  refined  methods 
which  will  be  considered  in  the  detailed  discussion,  are  very  close. 
The  American  Ephemeris  gives  the  values  of  X,  Y,  and  Z  for  every 
day  in  the  year,  and  from  these  data  the  values  of  their  first  and 
second  derivatives  can  be  found.  As  a  matter  of  fact  only  the 
first  derivatives  of  these  coordinates  will  be  required. 

Second  Step.  The  second  step  is  to  impose  the  condition  that  C 
moves  around  S  in  accordance  with  the  law  of  gravitation.  It 
will  be  assumed  that  C  is  not  sensibly  disturbed  by  the  attractions 
of  other  bodies.  Hence  its  coordinates  satisfy  the  differential 
equations 

'dzx  k2x 

dt2        ~  r2  ' 

fJ2ni  Tf^ll 

(<-r\  \  &  y        K  y 

*Wm    '.7' 

Off  =      "r3"' 
But  it  also  follows  from  the  relations  of  C,  E,  and  S  that 

x  =  PX  -  X, 

(8)  -{  y  =  PM  -  Y, 

z  =  pv  —  Z. 


Ill] 


OUTLINE   OF   THE  LAPLACIAN   METHOD. 


197 


On  substituting  these  expressions  for  x,  y,  and  z  in  equations  (7), 
they  become 


(9) 


(PM)"  -  Y 


(p.)"  -  Z"  - 


r3 

v  -  Z) 


But  since  E  also  revolves  around  S  in  accordance  with  the  law 
of  .gravitation,  it  follows  that 

y  —  _  *  X 
R*  > 

1.2V 

"V" 

'  & ' 


7n 

VBT- 


Therefore  equations  (9)  become 
Xp"  +  2X'p'+  |\"  + 

(10)   - 


p  =  -  VX 


The  unknown  quantities  in  these  equations  are  p",  p',  p,  and  r, 
the  first  three  of  which  enter  linearly. 

Third  Step.  The  third  step  is  to  determine  the  distance  of  C 
from  E  and  S  by  means  of  equations  (10)  and  a  geometrical 
condition  which  the  three  bodies  must  satisfy.  In  order  to  solve 
equations  (10)  for  p,  let 


(ID       D  = 


\      \'  \"  _i_ 

A,       A  ,  AT  ~3~ 

/  \  /J      I  y 

M,    M,  x   +  ^ 


X,     X',     X" 


"',  "" 


The  second  form  of  the  determinant  D  is  obtained  by  multiplying 


198 


OUTLINE    OF   THE    LAPLACIAN   METHOD. 


[Ill 


k2 
the  first  column  by  -5  and  subtracting  the  product  from  the  third 

column.     The  determinant  which  is  obtained  by  replacing  the 
elements  of  the  third  column  of  D  by  the  right  member  of  (10) 


will  also  be  needed.     If  the  common  factor  I  -^ 
this  determinant  is 


-^  --  - 


is  omitted, 


(12) 


X 


v  , 


The  determinants  D  and  DI  involve  only  known  quantities. 
The  solution  of  equations  (10)  for  p  is 

(13)  P  =  \ 

To  this  equation  in  the  two  unknown  quantities  p  and  r  must  be 
added  the  equation 

(14)  r2  =  p2  +  R2  -  2PR  cos  ^, 

which  expresses  the  fact  that  the  three  bodies  C,  S,  and  E  form  a 
triangle.  The  angle  ^  is  the  angle  at  E  between  R  and  p,  and 
this  equation  also  has  only  the  unknowns  p  and  r.  The  problem 
of  solving  (13)  and  (14)  for  p  and  r  is  that  which  constitutes  the 
third  step.  The  solution  of  this  problem  gives  the  coordinates 
of  C  by  means  of  equations  (8)  which  involve  only  p  as  an  unknown. 
Fourth  Step.  The  fourth  step  is  the  determination  of  the 
components  of  velocity  of  C.  It  follows  from  (8)  that 


= 


(15) 


y'  =  pV  +  P/*'  -  Y', 
=  p'v  +  Pvr  -  Z'. 

The  only  unknown  in  the  right  members  of  these  equations  is  p' 
which  can  be  determined  from  (10).     The  expression  for  it  is 


(16) 


,'-  -Mi.   11 

r2D[Rs      r3]' 


Z>«  =  - 


X,     X,     X" 
M,     Y,     IL" 

V,        Z, 

Therefore  xf,  y',  and  z'  become  known. 


v" 


112] 


OUTLINE   OF  THE   GAUSSIAN   METHOD. 


199 


Fifth  Step.  The  fifth  and  last  step  is  to  determine  the  elements 
of  the  orbit  from  the  position  and  components  of  velocity  of  the 
body.  This  is  the  problem  which  was  solved  in  chap.  v. 

112.  Outline  of  the  Gaussian  Method  of  Determining  an  Orbit. 

First  Step.  The  first  step  in  the  Gaussian  method  is  to  impose 
the  condition  that  C  moves  in  a  plane  passing  through  S.  Since 
S  is  the  origin  for  the  coordinates  x,  y,  and  z,  this  condition  is 

Axi  +  %i  +  Czi  =  0, 
Ax2  +  By*  +  Cz2  =  0, 
Axs  +  Bys  +  Cz3  =  0, 

where  A,  B,  C  are  constants  which  depend  upon  the  position  of 
the  plane  of  motion.  The  result  of  eliminating  the  unknown 
constants  A}  B,  and  C  is  the  equation 


(17) 


Xz, 


2/i, 


2/3, 


=  0. 


The  determinant  (17)  can  be  expanded  with  respect  to  the 
elements  of  the  three  columns  giving  the  three  equations 

'  (2/223  —  Z22/s)Xi  —    (2/123  —  Zi2/s)x2  +   (2/l22  —  Zi2/2)X3  =  0, 

(18)  -    (X2Z3  -  22X3)2/1   -    (XiZ3  -  21X3)2/2  +   (Xi22  -  2iX2)2/3   =   0, 

(x22/3  -  2/2^3)21  -  (xi2/3  -  2/1^3)22  +  (xii/2  -  2/1^2)23  =  0. 

Evidently  these  three  equations  are  but  different  forms  of  the  same 
one;  but  when  the  nine  parentheses  are  determined  from  additional 
principles  and  xi,  x2,  •  •  •  are  expressed  in  terms  of  the  geocentric 
coordinates  by  (8),  they  become  independent  in  the  unknowns 
pi,  Pz,  and  p3.  The  parentheses  are  the  projections  of  twice 
the  triangles  formed  by  S  and  the  positions  of  C  taken  in  twos 
upon  the  three  fundamental  planes.  Since  in  each  equation  the 
three  areas  are  projected  upon  the  same  plane  the  triangles 
themselves  can  be  used  instead  of  their  projections.  If  [1,  2], 
[1,  3],  and  [2,  3]  represent  the  triangles  formed  by  S  and  C  ai  the 
times  tit*,  Ms,  and  Ms  respectively,  equations  (18)  become 

[2,  3]xi  —  [1,  3]x2  +  [1,  2]x3  =  0, 

(19)  -(  [2,  3]  2/1  -  [1,  3]  i/2  +  [1,  2]  2/3  =  0, 

[2,  3]  2l  -  [1,3]  22 +  [1,2]  2,  =0. 


200 


OUTLINE   OF   THE   GAUSSIAN   METHOD. 


[112 


Second  Step.  The  second  step  consists  in  developing  the  ratios 
of  the  triangles  as  power  series  in  the  time-intervals.  This  is 
done  by  integrating  equations  (7)  as  power  series  in  the  time- 
intervals,  and  then  substituting  the  results  for  t  =  ti,  tz,  t3  in  the 
coefficients  of  (18)  or  (19).  Inasmuch  as  these  series  are  based 
upon  equations  (7)  the  condition  that  C  shall  move  about  S  in 
accordance  with  the  law  of  gravitation  has  been  imposed.  In 
order  not  to  prolong  the  discussion  at  this  point  (for  the  details 
see  Art.  127)  the  results  will  be  given  at  once.  For  the  purpose 
of  simplifying  the  writing,  let 


(20) 


In  this  notation  the  ratios  of  the  triangles  [2,  3]  and  [1,  2]  to  [1,  3] 
are  found  to  be 


-  t,)  =  03, 
k(ts  -  t2)  =  Oi, 


(21) 


ro    Ql 
l^>  "J  _ 


_l  - 

-         f          ~ 


[i, 

Uoi        a    r  1/32        /j  2 

>    *J    _   I3        1    _|_  i    ^2     —    #3 

Jl,  3]      02  L     r6       r23 


I 
J' 


Third  Step.  The  third  step  consists  in  developing  equations 
for  the  determination  of  pi,  p2,  and  p3.  The  results  of  substituting 
equations  (8)  and  (21)  in  (19)  are 


(22) 


0i 


01 


r,  i  i^2-0i; 

L1  +  6~7^ 


+  •• 

1022    - 


+ 


(x3P3  -  xt)  =  o, 


['+1*5* 


(Mipi  -  Fi)  - 


6       r23 


(M3P3    - 


-  0i5 


1022  - 
6       r23 


0, 


("3P3   -   Z8)    =   0. 


112]  OUTLINE    OF   THE   GAUSSIAN   METHOD.  201 

These  equations  involve  the  unknowns  pi,  p2,  PS,  and  r2,  the  first 
three  of  which  enter  linearly.  Since  r2  enters  only  as  it  is  multi- 
plied by  the  small  quantities  0i2,  022,  or  032,  it  might  be  supposed 
that  in  a  first  approximation  these  terms  could  be  neglected,  after 
which  pi,  p2,  and  p3  would  be  determined  by  linear  equations. 
A  detailed  discussion  of  the  determinants  which  are  involved 
shows,  however,  that  it  is  necessary  to  retain  the  terms  in  r2  even 
in  the  first  approximation. 

The  solution  of  equations  (22)  for  p2  has  the  form 

(23)  AP2  =  P  +  J, 

where  A  is  the  determinant  of  the  coefficients  of  pi,  p2,  and  p3, 
and  P  and  Q  are  functions  of  the  known  quantities  \i,  X2,  •  •  •  , 
Xi,  YI, 

Since  8,  E,  and  C  form  a  triangle  at  tz  the  quantities  p2  and  r2 
satisfy  the  equation 

(24)  7-22  =  p22  +  #22  -  2p2E2  cos  fa. 

The  solution  of  any  two  equations  of  (22)  for  pi  and  p3  in  terms 
of  p2  and  r2  has  the  form 


where  M,  PI,  P3  are  functions  of  known  quantities,  and  Q\  and  Q3 
involve  only  r2  as  an  unknown. 

Fourth  Step.  The  fourth  step  consists  in  determining  pi  and  ps. 
The  quantities  p2  and  r2  are  found  first  by  solving  (23)  and  (24), 
which  is  exactly  the  same  as  the  third  step  of  the  Laplacian 
method,  and  then  pi  and  p3  are  given  by  (25). 

Fifth  Step.  The  fifth  step  consists  in  determining  the  elements 
from  the  known  positions  of  C  at  the  times  t\  and  £3.  These  two 
positions  and  that  of  C  define  the  plane  of  the  orbit  without 
further  work.  Gauss  solved  the  problem  of  determining  the 
remaining  elements  by  developing  two  equations  involving  only 
two  unknowns.  One  equation  was  derived  from  the  ratio  of 
the  triangle  formed  by  S  and  C  at  ti  and  t3  to  the  area  of  the 
sector  contained  between  r\,  r3,  and  the  arc  of  the  orbit  described 
in  the  interval  tit*.  The  other  equation  was  derived  from  Kepler's 


202 


LAPLACIAN   METHOD    OF   DETERMINING    ORBITS. 


[113 


equation  at  the  epochs  ti  and  t^  The  formulas  are  complex, 
but  the  method  of  solving  the  two  equations  is  a  rapid  process 
of  successive  approximations.  After  the  equations  are  solved  the 
elements  are  uniquely  determined  without  any  trouble.  Later 
methods  have  been  devised  which  avoid  many  of  the  complexities 
of  that  due  to  Gauss. 

I.    THE  LAPLACIAN  METHOD  OF  DETERMINING  ORBITS. 

113.  Determination  of  the  First  and  Second  Derivatives  of  the 
Angular  Coordinates  from  Three  Observations.  It  was  found  in 
the  outline  [Art.  Ill]  of  this  method  of  determining  orbits  that 
the  first  and  second  derivatives  of  the  angular  coordinates,  or 
of  the  direction  cosines  X,  n,  and  v  will  be  required. 

Let  k(t  —  £o)  =  T  and  then  equations  (7)  become 


(26) 


_  _  £ 

~d?~  ~^' 

&y  =  _g. 

dr2  r3 ' 


tfz 
dr* 


£^ 

r3* 


a 

Suppose  x  =  x0)  y  =  yQ,  z 


z0, 


dx 


rfi/ 


$  =  *,'  at 


n\drn 


n\\drn0 


r  =  0.  The  solution  of  equations  (26)  can  be  expanded  as  power 
series  in  r  which  will  converge  if  the  value  of  r  is  not  too  great.* 
They  will  have  the  form 


(27) 


where  the  subscript  0  on  the  parentheses  indicates  that  the  deriva- 
tives are  taken  for  r  =  0.  The  second  derivatives  can  be  replaced 
by  the  right  members  of  (26)  for  r  =  0;  the  third  derivatives  can 
be  replaced  by  the  first  derivatives  of  the  right  members  of  (26), 
and  so  on.  All  the  derivatives  in  this  way  will  be  expressed  in 
terms  of  x0,  2/0,  ZD,  XQ',  yo',  and  z0'. 

*  For  the  determination  of  the  exact  realm  of  convergence  see  a  paper  by 
F.  R.  Moulton  in  The  Astronomical  Journal,  vol.  23  (1903). 


113]  FIRST  AND   SECOND   DERIVATIVES   OF   X,   /z,   V.  203 

It  is  important  to  know  for  how  great  intervals  the  series  (27) 
are  of  practical  value.  The  limits  are  smaller  the  smaller  the  peri- 
helion distance  and  the  greater  the  eccentricity,  and  moreover 
they  depend  upon  the  position  of  the  body  in  its  orbit  at  r  =  0. 
For  a  small  planet  whose  mean  distance  is  2.65,  which  is  about 
the  average  for  these  bodies,  and  the  eccentricity  of  whose  orbit 
does  not  exceed  0.4,  which  is  much  greater  than  that  of  most  of 
them,  the  series  (27)  always  converge  for  an  interval  of  less  than 
160  days.  If  the  orbit  is  a  parabola  whose  perihelion  distance  is 
unity  the  series  (27)  converge  if  the  interval  of  time  does  not 
exceed  54  days.  Of  course,  the  series  are  not  of  practical  value 
in  their  whole  range  of  convergence.  In  practice  in  the  case  of 
small  planets  an  interval  of  90  days  is  nearly  always  small  enough 
to  secure  rapid  convergence  of  (27),  and  in  the  case  of  the  orbits 
of  comets  20  days  is  rarely  too  great  an  interval. 

The  coordinates  of  the  earth  also  are  expansible  as  series  of  the 
form  of  (27),  and  the  rapid  convergence  holds  for  very  long 
intervals  because  of  the  small  eccentricity  of  the  earth's  orbit. 
Hence  it  follows  from  equations  (8)  that  p,  X,  ju>  and  v  can  be 
expanded  as  power  series  of  the  type  of  (27).  The  range  of 
usefulness  of  these  expansions  is  the  same  as  that  of  the  series 
for  x,  y,  and  z. 

It  will  be  sufficient  to  consider  the  series  for  X  because  those 
in  fjL  and  v  are  symmetrically  similar.  The  series  for  X  for  a 
general  value  of  r  and  for  n,  r2,  and  r3,  which  correspond  to 
h,  U,  and  £3  respectively,  are 

X    =  C0  +  CIT   +  C2r2    + 
(28)  Xi^o  +  cxn  +  c^2-!- 


X2    —    CQ     |      ClT2  -f-   C2T2     -p    *  *  *, 
X3    =    C0  +   CiT3  +   C2T32  -H    '."I 

where  Co,  Ci,  c2,  •  •  •  are  constants.  If  these  equations  are  termi- 
nated after  the  terms  of  the  second  degree  the  coefficients  Co,  Ci, 
and  c2  are  determined  in  terms  of  the  observed  quantities  Xi,  X2, 
and  X3,  and  the  time-intervals  n,  r2,  and  r3.  If  more  observations 
are  available  more  coefficients  can  be  determined;  the  number 
which  can  be  determined  equals  the  number  of  observations. 

The  simplest  way  of  expressing  X  in  terms  of  r  with  known 
coefficients  is  to  set  equal  to  zero  the  eliminant  of  1,  c0,  Ci,  and  c2 
in  (28),  which  is 


204 


LAPLACIAN   METHOD    OF   DETERMINING    ORBITS. 


[113 


(29) 


X,  1, 

Xi,  1, 

X2,  1, 

X3,  1, 


=  0. 


The  expansion  of  this  determinant  with  respect  to  the  elements  of 
the  first  column  is 


=   —  (r2  —  ri)(r3  —  TO)(TI  —  r3), 


and  where  A\,  A2,  and  A3  are  obtained  from  AQ  by  permuting  r 
with  TI,  T2,  and  r3  respectively.  The  determinant  Ao  is  distinct 
from  zero  if  TI,  T2,  and  T3  are  distinct.  Hence  equation  (30) 
becomes 

(r  -  r2)(r  -  T8)   ,  (r  -  r3)(r  -      x 


(30)                  A0X  -  Ai 

where 

1, 

Tl,        Ti2 

Ao  = 

1, 

T2,        Tl2 

1, 

T3,        T32 

X  = 


(31) 


(TI  —  TZ)(TI  —  r3)  (TZ  - 

(r  -  TI)(T  -  T2) 


X2 


, 


(r3  —  TI)(TS  — 


It  follows  from  the  form  of  (31)  that  this  equation  gives  X 
exactly  at  n,  TZ,  and  r3;  for  other  small  values  of  T  it  gives  X  ap- 
proximately. The  exact  value  of  X  is  given  by  an  infinite  series, 


l^SS- 

V 

~^ 

1 

"\ 

I 

\^ 

1 

1 

\ 

! 

t 

1 

<*                     / 

Fig.  31. 

the  first  equation  of  (28),  within  the  range  of  its  convergence. 
Geometrically  considered  this  series  defines  a  curve,  marked  C  in 
Fig.  31.  The  second  degree  polynomial  (31)  defines  another 


114] 


DERIVATIVES    FROM    FOUR    OBSERVATIONS. 


205 


curve  C2.  These  two  curves  intersect  at  TI,  T2,  and  T3,  but  in 
general  do  not  intersect  elsewhere.  For  small  values  of  T  the 
two  curves  nearly  coincide,  and  the  approximate  value  of  X  can 
be  found  from  the  polynomial  near  the  origin. 

The  first  and  second  derivatives  of  X  are  found  from  (31)  to  be 
given  approximately  by 

2T    -    (T2   +   T8)  2T    - 


(32) 


—    T2)(Tl    —    T3) 
2T-    I 


(r2  —  r3)(r2  —  TI) 
r2) 


—  TI)(TS  —  r2) 


X8| 


—    T2)(Tl    —    T3) 


(T2    —    T3)(T2    —    Tl) 


X2 


X2 


X3. 


TS  —  TI)(TS  —  r2) 
There  are  similar  expressions  in  ju  and  v. 

114.  Determination  of  the  Derivatives  from  more  than  Three 
Observations.  If  the  observations  were  perfectly  exact  and 
near  together,  the  more  there  were  available  the  more  exactly 
could  X  be  determined  for  small  values  of  T,  and  the  more  of  its 
derivatives  could  be  determined.  Suppose  there  are  four  obser- 
vations. Then  X  is  defined  by  a  third  degree  polynomial  analogous 
to  (31)  which  reduces  to  Xi,  X2,  X3,  and  X4  for  r  =  TI,  r2;  r3,  and  r4 
respectively.  The  explicit  expression  for  X  is 

(r  —  r2)(r  —  r3)(r  —  r4)    ^ 
TI  —  r2)(ri  —  TS)(TI  —  T4) 

(r  —  TS)(T  —  r4)(r  — 


(33) 


X=  + 


(TZ   —    T3)(T2    —    T4)(T2    —    TI 

(T  —  T4)(r  —  TI)(T  —  T2) 


TS   —    T4)(T3    — 

(T  —  TI)(T  — 


S    —    T2) 

—  T3) 


X2 


X3 


X4, 


(T4    —    Ti)(T4    —    T2)(T4    —    T3) 

from  which  the  first,  second,  and  third,  but  not  higher,  derivatives 
can  be  found. 

It  is  obvious  from  this  how  to  proceed  for  any  number  of  obser- 
vations. The  process  is  unique  and  does  not  become  excessively 
laborious  unless  the  number  of  observations  is  considerable.  The 
number  of  derivatives  which  can  be  determined,  at  least  approxi- 
mately, is  one  less  than  the  number  of  observations,  but  no 


206 


LAPLACIAN   METHOD   OF   DETERMINING   ORBITS. 


[115 


derivative  higher  than  the  third  will  in  any  case  be  used.  If  the 
observations  extend  over  a  long  period  so  that  the  convergence 
of  (28)  fails  or  becomes  slow  for  the  largest  values  of  r,  it  is  neces- 
sary to  omit  some  of  them  in  the  discussion.  Usually,  owing  to 
the  errors  in  the  observations,  four  or  five  will  give  X  and  its 
first  two  derivatives  as  accurately  as  any  greater  number. 

115.  The  Approximations  in  the  Determination  of  the  Values 
of  X,  M>  v  and  their  Derivatives.  In  the  applications  it  is  im- 
portant to  know  the  character  of  the  approximations  which  are 
made,  and  whether  all  the  quantities  employed  are  determined 
with  the  same  degree  of  accuracy.  It  is  obvious  no  exact  numerical 
answers  can  be  given  to  these  questions  because  the  orbits  under 
consideration  are  undetermined.  But  it  has  been  insisted  that 
the  values  of  r  must  not  be  too  great  in  order  that  the  series  (28) 
shall  converge  rapidly.  Consequently,  the  values  of  r  at  the 
times  of  the  observations  can  be  considered  as  small  quantities, 
and  the  degree  of  the  approximation  can  be  described  in  terms 
of  the  lowest  powers  of  the  T,  which  occur  in  the  neglected  terms. 
This  gives  a  definite  meaning  to  the  order  of  approximation,  and 
experience  shows  that  it  is  a  satisfactory  measure  of  the  accuracy 
of  the  results  when  the  time-intervals  are  limited  as  described 
in  Art.  113. 

Suppose  first  that  only  three  observations  have  been  made. 
The  approximations  in  the  determination  of  X  and  its  derivatives 
arise  from  the  fact  that  the  higher  terms  of  (28)  are  neglected. 
The  coefficients  c0,  Ci,  and  c2  are  determined  by 

Co  +  Cm  H-  C2r  i2  =  Xi  —  C3Ti3  —  C4Ti4  —    •  •  •, 


ClT2 
CiT3 


C2r  i 

C2T22    =    X2    -    C3T23    -    C4T24    - 


C2T32    =    X3    —    C3T3 


C4T3 


4    — 


The  errors  of  lowest  degree  in  the  TJ  come  from  neglecting  the 
terms  in  the  right  members  which  are  multiplied  by  the  unknown 
constant  c3.  Let  the  errors  be  denoted  by  Aco,  Aci,  and  Ac2. 
Then 

C4Tl4  +    •  • 


T3, 


T!2 

C3Ti3 

T22 

Aco  =  — 

C3T23 

T32 

C3T33 

Tl3 

=    —  C.3 

T23 

T33 

C4T24  + 
C4T34  + 
Tl,        Tl2 
T2,        T22 
T3,        T32 


—  C4 


Tl, 

Tl2 

• 

T2, 

T22 

o 

T3, 

T32 

Tl4, 

Tl,        Ti2 

T24, 

T2,        T22 

T34, 

T3,        T32 

116]  CHOICE    OF   THE    OKIGIN    OF   TIME.  207 

and  similar  expressions  for  Aci  and  Ac2.  These  determinants  are 
easily  reduced  by  the  elementary  rules  for  simplifying  deter- 
minants, and  it  is  found  that 

^Co  =  —  C3TlT2T3  —  C4TiT2T3(ri  +  T2  +  T3)  +  •  •  ', 
^Cl  =  +  C3(TlT2  +  T2T3  +  T3Tl) 

(35)    -  +  C4(ri  +  r2)(r2  +  r3)(r3  +  n)  +  •  •  *, 

Ac2  =  —  C3(ri  +  r2  +  r3) 

—    C4(Tl2  ~h   T22   ~|-   T32   -{~   TlT2   -f-   T2T3   -f-   T3Ti)    + 

It  follows  from  these  equations  that  c0,  Ci,  and  c2  are  determined 
up  to  the  third,  second,  and  first  orders  respectively. 

Now  consider  the  first  equation  of  (28).  Since  Ci  is  multiplied 
by  r  and  c2  by  r2,  each  of  ^he  first  three  terms  in  the  series  for  X  is 
determined  up  to  the  third  order  in  the  r/.  On  taking  the  first  and 
second  derivatives,  it  is  seen  that  X'  and  X"  are  determined  up  to 
the  second  and  first  orders  respectively.  Consequently,  X'in 
general  is  determined  by  the  first  terms  of  (28)  more  accurately 
than  its  first  derivative,  and  its  first  derivative  in  general  is 
determined  more  accurately  than  its  second  derivative.  These 
facts  must  be  remembered  in  the  applications. 

116.  Choice  of  the  Origin  of  Time.  The  origin  of  time  has 
not  been  specified  as  yet  except  that  it  has  been  supposed  that  it  is 
near  the  dates  of  the  observations  so  that  n,  r2,  and  r3  will  be 
small.  Any  epoch  fa  which  satisfies  this  condition  can  be  used 
as  an  origin,  and  the  problem  at  once  arises  of  determining  what 
one  is  most  advantageous. 

The  choice  of  the  origin  of  time  which  has  been  universally  made 
is  the  date  of  the  second  observation.  That  is,  fa  =  fa  and  there- 
fore r2  =  0.  The  value  of  X  is  exactly  known  at  r  =  r2  =  0,  and 
the  derivative  of  X  at  t  =  fa  is 

X2'  =  Ci  -j-  2c2r2  -}-  •  •  •  =  Cit 

which  is  subject  to  the  error  Aci,  which,  by  (35),  is  in  this  case 
c3T3ri.  And  similarly,  the  error  in  X2"  is  Ac2  =  —  c3[ri  +  r3]. 
The  error  in  X2'  is  of  the  second  order  while  that  in  X2"  is  of  the 
first  order.  In  general,  an  error  of  the  first  order  is  more  serious 
than  one  of  the  second  order.  But  it  should  be  noticed  that 
when  £o  =  fa  the  quantities  n  and  r3  are  opposite  in  sign;  and  if 
the  intervals  between  the  successive  observations  are  equal, 
TI  H-  r3  =  0  and  the  error  in  X2"  is  also  of  the  second  order.  Con- 


208  LAPLACIAN   METHOD   OF   DETERMINING    ORBITS.  [117 

sequently,  when  to  =  £2  it  is  advantageous  to  have  the  successive 
observations  separated  by  as  nearly  equal  time-intervals  as 
possible.  But  unfavorable  weather  and  other  circumstances 
generally  cause  the  observations  to  be  unequally  spaced. 

Suppose  the  epoch  of  the  first  observation  is  taken  as  the  origin 
of  time.  The  quantity  Xi  is  exactly  known*  The  error  in  X/  is 
Aci  =  c3T2T3,  which  is  of  the  second  order  as  before,  but  is  approxi- 
mately twice  as  great  numerically  as  that  in  X2'  because  r3  now 
represents  k  times  the  whole  interval  between  the  first  and  third 
observations.  The  error  in  X/'  is  Ac2  =  —  C3(r2  +  r3)  which 
is  much  larger  than  before  because  r3  now  depends  on  the  whole 
interval  covered  by  the  observations,  and  because  r2  and  r3  in 
this  case  are  both  positive.  It  follows  from  this  that  it  is  not 
advantageous  to  use  the  time  of  the  first  observation  as  the  origin 
of  time;  and  for  similar  reasons  the  epoch  of  the  third  observation 
is  to  be  rejected. 

The  question  now  arises  what  should  be  taken  for  the  origin 
of  time  when  the  epoch  of  the  second  observation  is  not  midway 
between  those  of  the  other  two.  Since  in  general  the  error  in  X  is 
only  of  the  third  order  and  that  in  X'  is  only  of  the  second,  while 
X"  is  subject  to  an  error  of  tne  first  order,  it  is  clear  that  the  origin 
of  time  should  be  so  chosen,  if  possible,  as  to  make  the  first  order 
error  in  X"  vanish.  It  follows  from  the  second  equation  of  (35) 
that  this  result  will  be  secured  if 

fri  +  r2  +  r3  =  k(ti  -  to)  +  kfa  -  t0).+  k(ts  -  t0)  =  0, 
(36)    4 

[     whence   to  =  ^  (ti  +  £2  +  £3). 

The  best  choice  of  the  origin  of  time  is  therefore  given  by  the 
second  of  (36),  and  this  value  of  t0  becomes  the  date  of  the  second 
observation  when  the  successive  observations  are  equally  distant 
from  one  another.  With  this  choice  of  to  the  errors  in  X'  and  X" 
are  of  the  second  order,  while  X  is  known  up  to  the  third  order. 

117.  The  Approximations  when  there  are  Four  Observations. 

When  there  are  four  observations  the  equations  which  correspond 
to  the  last  three  of  (28)  are 

CO  +  CiTi  +  C2Ti2  +  C3Ti3  =  Xl  -  C4Ti4  H , 

J  C°  ~*~  ClT2  +  C2T22  +  C3T23  =  ^2  —  C4T24  H , 

(o7) 

O  +   CiT3   +   C2T32   +   C3T33    =    X3    —   C4T34   -I , 

I  C0  +   CiT4   -f   C2T42    +    C3T43    =    X4   —    C3T44    ~\ . 


117]   APPROXIMATIONS  WHEN  THERE  ARE  FOUR  OBSERVATIONS.     209 


The  determinant  of  the  coefficients  of  Co,  Ci,  c2,  and  c3  is 


5  = 


T32, 
T42, 


T2° 
T33 

T43 


=  (TZ  —  TI)(TS  —  Ti)(r4  —  Ti)(r8  —  r2) 
X  (r4  —  T2)(T4  —  T3), 


which  is  not  zero  since  the  dates  of  the  observations  are  distinct. 
The  errors  of  lowest  order  in  c0,  Ci,  c2,  and  c3  are  determined 
from  (37);  when  only  the  first  terms  in  the  right  members  are 
known  they  contain  c4  as  a  factor.  Let  these  errors  be  represented 
by  Ac0,  Aci,  Ac2,  and  Ac3;  their  orders  in  the  TJ  are  required.  The 
expression  for  Aco  is 


Ac0  = 


—  C4 


Tl4, 
T24, 
T34, 
T44, 


T3, 


IV, 

T32, 
T42, 


Tl 


T2 


T3 


T4 


When  the  factors  TI,  TZ,  r3,  and  r4  are  removed  from  this  deter- 
minant it  is  identical  with  5  except  the  columns  are  permuted. 
Three  permutations  of  columns  bring  it  to  the  form  of  6;  hence 

(38) 


AC0    = 


C4TiT2T3T4. 


1, 

Tl4, 

Tl2, 

Tl3 

-  c4 

1, 

T24, 

T22, 

T23 

0 

1, 

T34, 

T32, 

T33 

1, 

T44, 

T42, 

3 

The  expression  for  Aci  is 


Ac,  = 


If  TZ  is  put  equal  to  TI  in  this  determinant  it  vanishes  because  then 
two  lines  become  the  same.  Therefore  it  is  divisible  by  TZ  —  TI. 
Similarly,  it  is  divisible  by  r3  —  TI,  T4  —  TI,  T3  —  T2,  T4  —  T2, 
and  T4  —  T3;  that  is,  it  is  divisible  by  6.  All  the  elements  of  each 
column  are  of  the  same  degree;  and  since  every  term  of  the  ex- 
pansion of  a  determinant  has  a  factor  from  each  column,  the  terms 
of  the  expansion  are  all  of  the  same  degree.  The  degree  of  this 
determinant  is  nine,  because  this  is  the  sum  of  the  degrees  of  its 
columns.  Hence  Aci  is  of  the  third  degree  because  d  is  of  the 
sixth  degree.  Moreover,  it  is  symmetrical  in  TI,  •  •  • ,  T4  because 
both  8  and  the  numerator  determinant  are  symmetrical  in  these 
quantities.  Each  term  of  the  expansion  contains  TJ  only  to  the 
15 


210  LAPLACIAN   METHOD   OF  DETERMINING   ORBITS.  [117 

first  degree  because  77  occurs  in  the  numerator  determinant  to 
the  fourth  degree  as  the  highest,  and  in  d  to  the  third  degree.  The 
numerical  coefficient  of  each  term  in  the  expansion  is  the  same, 
because  of  the  symmetry,  and  it  can  be  determined  by  the  con- 
sideration of  a  single  term.  It  is  found  by  considering  the  product 
of  the  main  diagonal  elements  that  it  is  -}-  1.  Analogous  dis- 
cussions can  be  made  for  Ac2  and  Ac3,  and  it  is  found  in  this  way 
that 

f  ACl  =  —  C4[TiT2T3  +  T2T3T4  +  T3T4Tl  +  T4TiT2], 
(39)     -I  AC2  =  +  C4[TiT2  +  TiT3  +  TiT4  +  T2T3  +  T2T4  +  T3T4], 

[Ac3  =   -  C4[ri  +  r2  +  r3  +  rj. 

It  follows  from  (38)  and  (39)  that  when  there  are  four  obser- 
vations X,  X',  X",  and  X"'  are  determined  up  to  small  quantities 
of  the  fourth,  third,  second,  and  first  order  respectively.  Ordi- 
narily X'"  is  not  needed,  though  it  becomes  useful  when  the  solution 
is  double,  as  it  may  be,  in  determining  which  of  them  belongs  to 
the  physical  problem.  In  this  latter  case  it  is  advantageous  to 
make  Ac3  vanish  by  determining  tQ  so  that 

f  TI  +  r2  +  T3  +  r4  =  0,     whence 
(40)  4 

1*0    =    i01   +   *2   +   *3   +   «• 

If  the  solution  of  the  problem  is  made  to  depend  only  on  X,  X', 
and  X",  it  is  most  advantageous  to  choose  t0  so  that  Ac2  shall 
vanish,  for  then  all  the  quantities  are  determined  up  to  the  third 
order.  This  condition  becomes 


(41) 


TlT2  +  TiT3  +  TiT4  +  T2T3  +  T2T4  +  T3T4  =  0, 

W  -  30!  +  t*  +  t,  +  «*o  +  W2 

+  Ms  +  tit*  +  *2Z3  +  *2*4  +  t,U  =  0. 


The  values  of  Zo  determined  by  this  quadratic  equation  are  of 
no  practical  value  unless  they  are  real.  The  discriminant  of  the 
quadratic  is 


=  H  -  30!  -  ttf  +  3(*i  -  t.Y  +  30i  -  UY 

+  302  -  *3)2  +  302  -  ttf  +  303  -  *4)2  >  0. 
Therefore  the  solutions  are  always  real,  and  are  explicitly 


= 


118] 


THE   FUNDAMENTAL   EQUATIONS. 


211 


In  order  to  get  a  concrete  idea  of  the  nature  of  the  results 
suppose  the  intervals  between  the  successive  observations  are 
equal  to  T.  Then  (42)  gives 

(43)  tQ  =  i(«i  +  U  +  t,  +  U)  ±  t  Vl5  T. 

The  first  term  on  the  right  is  the  mean  epoch  of  the  observations, 
and  the  two  values  of  tQ  are  at  the  distance  J  Vl5  T  either  side  of 
this  time.  Since  the  interval  between  the  mean  epoch  and 
tz  or  £3  is  f  71,  it  follows  that  t0  is  between  ti  and  tz  and  distant 
($Vl5  —  %)T  =  %T  approximately  from  £2,  or  symmetrically  situ- 
ated between  t3  and  U.  In  practice  it  will  be  most  convenient 
to  choose  IQ  =  fa  or  t0  —  ts,  for  then  X  is  given  exactly,  the  coef- 
ficients of  (33)  are  as  simple  as  possible,  and  (41)  is  nearly  satisfied. 
The  discussion  when  there  are  five  or  more  observations  can  be 
carried  out  in  a  similar  manner.  For  each  additional  observation 
one  additional  coefficient  in  the  series  (28)  can  be  determined, 
and  those  which  were  determined  previously  become  known  to 
one  order  higher  in  the  T/.  In  each  case  one  additional  order  of 
accuracy  in  the  determination  of  X"  can  be  secured  by  properly 
selecting  Z0,  but  it  is  simplest  to  let  U  equal  the  date  of  the  obser- 
vation which  is  nearest  the  mean  epoch  of  all  of  the  observations. 

118.  The  Fundamental  Equations.  The  fundamental  equations 
of  the  method  of  Laplace  are  (10),  where  X,  n,  v,  X',  M'>  v',  X",  /*",  /' 
are  given  by  (31)  and  (32)  and  corresponding  equations  in  p  and  v. 
The  solution  of  equations  (10)  for  p,  p',  and  p"  is 


(44) 


where 


(45)  - 


2D    R3      r3 


-I[n    -^ 
~I>L  r8 


X" 


v" 


Dl  =  - 


D  -  - 


X,     X,  X" 

M,  y,  /•" 

JV  // 

V«          ^v  •  J^ 


x, 

X',     Z 

M, 

M',     Y    , 

*i 

/,    z 

X", 

X',     X" 

r, 

M',     M" 

ry 

/        // 

1 

212 


LAPLACIAN   METHOD   OF   DETERMINING   ORBITS. 


[119 


These  determinants  are  subject  to  small  errors  because  of  the 
fact  that  the  higher  terms  of  equations  (28)  have  been  neglected. 
After  p  and  p  have  been  approximately  determined  corrections 
can  be  made  for  these  omissions.  The  determinants  are  also  sub- 
ject to  small  errors  because  they  have  been  developed  under  the 
tacit  assumption  that  the  observations  were  made  from  the 
position  of  the  center  of  the  earth  instead  of  from  one  or  more 
points  on  its  surface.  After  the  approximate  distances  have 
been  determined  the  observations  can  be  corrected  for  the  effects 
of  the  observer's  position  on  the  surface  of  the  earth. 

119.  The  Equations  for  the  Determination  of  r  and  p.  Con- 
sider the  triangle  formed  by  S,  E,  and  C.  Let  ^  represent  the 
angle  at  E  and  <p  that  at  C.  Then  it  follows  that 


(46) 


R 


p  = 


JR+Tn.+  Sr, 

D  sin  (^  +  <p)          i 

ti ; «          / 

sin  & 


sin 


When  equations  (46)  are  substituted  in  the  first  equation  of  (44) 
the  result  is 


R  sin  *  cos  v  +     fi  cos  *  - 


[fi 


sn      = 


In  order  to  simplify  this  expression  let 

'N  sin  m  =  R  sin 


(47) 


N  cos  m  =  R  cos  i/"  — 
M 


DR3' 
-  NDR*  sin3 


119]         EQUATIONS  FOR  THE   DETERMINATION   OF  r  AND   p.  213 


where  the  sign  of  N  will  be  so  cnosen  that  M  shall  be  positive. 
With  this  determination  of  the  sign  of  N  the  first  two  equations 
of  (47)  uniquely  determine  N  and  m,  and  the  equation  in  <p  becomes 
simply 

(48)  sin4  <p  =  M  sin  (<p  -f  m). 

The  quantities  M  and  m  are  known  and  M  is  positive. 

Now  consider  the  solution  of  (48)  for  <p.  Since  p  =  0,  r  =  R 
is  a  solution  of  the  problem,  it  follows  from  (48)  that  <p  =  TT  —  \f/ 
is  a  solution  of  (48).  This  solution  belongs  to  the  position  of  the 
observer  and  is  to  be  rejected.  It  follows  from  Fig.  32  that  the  <p 
belonging  to  the  physical  problem,  which  must  exist  if  the  compu- 
tation is  made  from  good  observations,  satisfies  the  inequality 

(49)  <p   <   7T   -    ^. 

The  solutions  of  (48)  are  the  intersections  of  the  curves  defined  by 
the  equations 


(50) 


2/2  =  M  sin(<?  +  m). 


For  m  negative  and  near  zero  and  M  somewhat  less  than  unity 
these  curves  have  the  relation  shown  in  Fig.  33. 
y 


Fig.  33. 

Consider  first  the  case  where  -=^  is  positive.     Since  both  p  and  r 

must  be  positive,  it  follows  from  the  first  of  (44)  that  in  this  case 
r  >  R.  Since  ^  is  less  than  180°,  it  follows  from  (47)  that  N  is 
negative,  and  that  m  is  in  the  third  or  fourth  quadrant. 

In  case  m  is  in  the  fourth  quadrant  the  ascending  branch  of 
the  curve  yz  crosses  the  <p-axis  in  the  first  quadrant,  and,  if  M  <  1, 
the  relations  of  the  curves  are  as  indicated  in  Fig.  33.  If  m  is 


214 


LAPLACIAN   METHOD   OF   DETERMINING   ORBITS. 


[119 


near  180°  there  are  three  solutions,  <pi,  <pz,  and  <p3,  one  of  which 
is  TT  —  \f/  and  belongs  to  the  position  of  the  observer.  If  (p3  =  7r  —  ^} 
both  <pi  and  (pz  fulfill  all  the  conditions  of  the  problem  and  it  can 
not  be  determined-which  belongs  to  the  orbit  of  the  observed  body 
without  additional  information.  However,  it  might  happen  that 
(Pi  would  give  so  great  values  of  r  and  p  that  it  would  be  known 
from  practical  observational  considerations  that  the  body  would 
be  invisible;  it  would  be  known  in  this  case  that  <p2,  which  would 
give  a  smaller  r,  belongs  to  the  physical  problem.  If  <p2  =  TT  —  \l/, 
it  follows  from  (49)  that  <pi  belongs  to  the  problem.  The  case 
(Pi  =  TT  —  \f/  cannot  occur  for  then  the  physical  problem  could 
have  no  solution.  If,  for  a  fixed  M,  the  ascending  branch  of  the 
curve  2/2  moves  to  the  right  the  roots  <pi  and  <p2  approach  coinci- 
dence; and  as  it  moves  farther  to  the  right  <p3  alone  remains  real. 
This  case,  which  corresponds  to  m  far  from  180°  in  the  fourth 
quadrant  or  in  the  third  quadrant,  cannot  arise,  for  then  the 

problem  would  have  no  solution.     Therefore,  if  -=£  is  positive,  then 

r  >  R,  m  is  in  the  fourth  quadrant,  and  there  are  one  or  two  possible 
solutions  of  the  physical  problem  according  as  <p2  or  <p3  equals  TT  —  \J/. 

Now  suppose  -— ^  is  negative.     In  this  case  r  <  R  and  m  is  in 

the  first  or  second  quadrant.  If  m  is  in  the  first  quadrant  the 
descending  branch  of  the  curve  yz  crosses  the  ^>-axis  in  the  second 


<t>t    TT_         <j>2  $3  * 

Fig.  34. 

quadrant,  and  for  a  small  m  and  M  <  1  the  relations  are  as  shown 
in  Fig  34.  In  this  case  the  solution  of  the  problem  is  unique  or 
double  according  as  <?2  or  <pz  equals  TT  —  \f/.  If  m  is  in  the  second 
quadrant  the  descending  branch  of  the  curve  yz  crosses  the  <p-axis 


120]  THE    CONDITION   FOR  A   UNIQUE   SOLUTION.  215 

in  the  first  quadrant,  <?2  and  <pa  are  not  real,  and  the  problem  has 
no  solution.  Therefore,  if  -~  is  negative,  then  r  <  R,  misin  the 
first  quadrant,  and  there  are  one  or  two  possible  solutions  of  the 
physical  problem  according  as  #2  or  <p3  equals  TT  —  ^. 

120.  The  Condition  for  a  Unique  Solution.  The  solution  of 
the  physical  problem  is  unique  whether  -^  is  positive  or  negative 
if  <pz  =  TT  —  \j/j  and  otherwise  it  is  double.  Suppose  p  =  TT  —  \f/  +  e, 
where  e  is  a  small  positive  number.  When  —  is  positive,  it  is 
seen  from  Fig.  33  that  if  <?2  =  TT  -  ^  the  difference  yi  -  yz  is 
positive  for  p  =  <?%  +  e;  and,  when  —  ^  is  negative,  it  is  seen  from 

Fig.  34  that  yi  —  yz  is  negative  for  v  =  ^  +  e  =  TT  —  $  +  e. 

It  follows  from  (50)  that  yi  and  y2  can  be  expanded  as  power 
series  in  e  when  <p  =  TT  —  $  +  e.  The  first  two  terms  of  the 
difference  are 

yi  —  2/2  =  [sin4  (TT  —  ^)  —  M  sin  (TT  —  \j/  +  m)] 
(51)  +  [4  sin3  (TT  -  ^)  COS  (TT  -  ^) 

—  M  cos  (TT  —  ^  +  m)]<=  +  •  •  •  . 

The  term  independent  of  e  is  zero  because  <p  =  TT  —  $  is  a  solution 
of  (48).     A  reduction  of  the  coefficient  of  e  by  equations  (47) 

and  (48)  gives 

MR  3Di 


Therefore  the  condition  that  the  solution  of  the  physical  problem 
shall  be  unique  is 


•f          J-S1 

(52) 

'  <  0    if     ^ 


This  function  is  completely  determined  by  the  observations,  and 
consequently  it  is  known  without  solving  (48)  whether  the  solution 
of  the  problem  is  unique  or  double. 
The  limit  of  the  inequalities  (52)  is 

r 

(53)  1 


216 


LAPLACIAN   METHOD   OF   DETERMINING    ORBITS. 


[120 


On   eliminating   cos  \f/   and   — ^  by  the  first  equations  of  (44)  and 
(46),  it  is  found  that 
(54)  P2  =  1 


2  R*      5 


The  minimum  value  of  the  right  member  of  this  equation,  con- 
sidered as  a  function  of  r,  is  zero;  therefore  for  each  value  of  r 
there  is  a  unique  positive  value  of  p.  All  points  defined  by  pairs 
of  values  of  r  and  p  which  satisfy  (54)  are  on  the  boundary  of  the 
regions  where  the  inequalities  (52)  are  satisfied.  These  boundary 
surfaces  are  evidently  surfaces  of  revolution  around  the  line 
joining  the  earth  and  the  sun.  The  section  of  these  surfaces  by  a 
plane  through  the  line  SE  is  shown  in  Fig.  35.* 


Fig.  35. 

The  surfaces  defined  by  (54)  divide  space  into  four  parts,  two 
of  which  in  the  diagram  are  shaded,  and  two  of  which  are  plain. 
The  function  (52)  has  the  same  sign  throughout  each  of  these 
regions  and  changes  sign  when  the  boundary  surface  is  crossed 


*  This  figure  was  first  given  by  Charlier,  Meddelande  fran  Lunds  Observa- 
torium,  No.  45. 


120]  THE    CONDITION   FOR  A   UNIQUE   SOLUTION.  217 

at  any  ordinary  point.  This  is  a  special  case  of  a  general  propo- 
sition which  will  be  proved. 

Suppose  XQ,  2/0,  ZQ  is  an  ordinary  point  on  the  surface  defined  by 
F(x,  y,  z)  =  0.  Consider  the  value  of  F  at  XQ  +  Ax,  yQ  +  Ay, 
ZQ  -f  Az,  where  Ax,  Ay,  and  Az  are  small.  The  value  of  the  function 
at  this  point  is 

F(XQ  +  Ax,  2/0  +  Ay,  Z0  +  Az) 

^+  —       -4-— A    +—  A    4- 
'  dx  dy  dz 

The  first  term  in  the  right  member  of  this  equation  is  zero  because 
XQ}  2/0,  ZQ  is  on  the  surface.  Now  suppose  the  point  XQ  +  Ax,  -  •  • 
is  on  the  perpendicular  to  the  surface  at  XQ,  y0,  ZQ.  Then 

dF 


Ax 


dF 


Ay  = 

I    /     «.  -n  \     n  /    n  T-f  \     n  *     *v  Try   \    O 


dF 

Az  = 


dFV         dF\*' 


where  p  is  the  distance  from  XQ,  2/0,  ZQ  to  XQ  +  Ax,  2/0  +  Ai/,  ZQ  +  Az, 
because  the  factors  by  which  p  is  multiplied  are  the  direction 
cosines  of  the  normal  to  the  surface.  On  one  side  of  the  surface  p 
is  positive,  and  on  the  other  side  it  is  negative.  The  expression 
for  the  value  of  the  function  F  at  the  point  XQ  +  Ax,  •  •  •  becomes 

F(xQ  +  Ax,  2/0  +  Ay,  ZQ  +  Az) 


For  p  very  small  the  sign  of  the  function  is  determined  by  the 
sign  of  the  first  term  on  the  right  whose  coefficient  is  not  zero. 
Since  XQ,  2/0,  ZQ  is  by  hypothesis  an  ordinary  point  of  the  surface, 
not  all  of  the  first  partial  derivatives  of  F  are  zero,  and  conse- 
quently the  sign  of  the  function  changes  with  the  change  of  sign 


218  LAPLACIAN   METHOD    OF   DETERMINING    ORBITS.  [121 

of  p.  That  is,  the  function  changes  sign  when  the  surface  for 
which  it  is  zero  is  crossed;  and  it  does  not  change  sign  at  any 
other  finite  point  because  the  function  is  continuous. 

In  order  to  find  in  which  of  the  four  regions  of  Fig.  35  the  solu- 
tion is  unique,  and  in  which  it  is  double,  consider  a  point  on  the 
line  SE  to  the  left  of  E.  At  such  a  point  r  =  p  +  R,  \I/  =  TT,  and 
it  follows  that 


1  DR^  DR* 

which  is  clearly  negative  for  p  very  large.  Since  in  this  case 
r  >  R  it  follows  that  -~  >  0,  N  <  0  and  the  first  inequality 

of  (52)  is  the  one  under  consideration.  Since  the  inequality  is 
satisfied  the  solution  of  the  problem  is  unique  if  the  observed 
body  is  in  the  unshaded  area  to  the  left  of  E.  If  the  surface  is 
crossed  into  the  larger  shaded  area  at  a  point  for  which  r  >  R 
the  function  changes  sign  while  the  sign  of  N  is  unchanged.  Then 
the  first  inequality  of  (52)  is  not  satisfied  and  the  solution  of  the 
physical  problem  is  double.  In  this  region  the  function  (53)  is 
positive  and  N  is  negative.  If  the  surface  is  crossed  into  the 
smaller  unshaded  area  the  function  (53)  becomes  negative,  N 
becomes  positive,  and  the  second  inequality  of  (52),  which  is  now 
in  question,  is  satisfied.  Therefore  the  solution  is  unique  in  this 
unshaded  area.  It  is  shown  similarly  that  it  is  double  in  the 
smaller  shaded  area. 

121.  Use  of  a  Fourth  Observation  in  Case  of  a  Double  Solution. 

Suppose  <p3  =  TT  —  i/'  so  that  there  are  two  solutions  of  (48)  which 
correspond  to  the  conditions  of  the  physical  problem.  One 
method  of  determining  which  solution  actually  belongs  to  the 
physical  problem,  in  case  there  are  four  observations,  is  obviously 
to  develop  (48),  using  the  fourth  observation  instead  of  one  of 
the  original  three.  In  general,  this  will  make  the  result  unique. 
A  better  method  of  resolving  the  ambiguous  case  can  be  devel- 
oped from  equations  (44).  Eliminate  r  from  the  second  and 
third  equations  of  (44)  by  means  of  the  first.  The  results  are 


122]  THE   LIMITS   ON   m   AND   M.  219 

The  derivative  of  the  first  of  these  equations  is 

P"  r  P'p  +  PPf  =  (Pf  +  P2)P, 

which  equated  to  the  right  member  of  the  second  equation  gives 
(55)  D,  -  |i-  +  Dp  =  Z),(P'  +  P»). 

Since  this  equation  is  linear  p  is  uniquely  determined  unless  D  is 
zero.  The  determinant  D  will  be  examined  in  Art.  124.  Equa- 
tion (55)  must  be  based  upon  not  less  than  four  observations,  for 
P'  involves  X"',  /JL'",  and  v'"  which  cannot  be  determined,  even 
approximately,  from  three  observations. 

122.  The  Limits  on  m  and  M.  In  an  actual  problem  of  the 
determination  of  an  orbit  the  constants  m  and  M  are  subject  to 
the  condition  that  equation  (48)  shall  have  three  real  roots  between 
0  and  TT.  The  limits  imposed  by  this  condition  can  be  determined 
from  the  conditions  that  it  shall  have  double  roots;  for,  suppose 
M  is  fixed  and  that  m  varies.  In  the  first  case,  represented  in 
Fig.  33,  there  are  three  real  solutions  of  (48)  until,  the  curve  yz 
moving  to  the  right,  <pi  and  <?2  become  equal;  and  in  the  second 
case,  represented  in  Fig.  34,  there  are  three  real  solutions  of  (48) 
until,  the  curve  yz  moving  to  the  left,  <p2  and  <p3  become  equal. 
The  two  cases  are  not  essentially  different  for  <p\  in  the  first  case 
corresponds  exactly  to  <p3  in  the  second.  Similarly,  if  m  remains 
fixed  and  M,  starting  from  a  small  value,  increases  there  are  three 
real  solutions  of  (56)  until  either  <p2  and  <p$  or  <pi  and  <p2,  in  the 
first  and  second  cases  respectively,  become  equal.  When  the 
limits  are  passed  for  which  two  values  of  <p  which  satisfy  (48) 
are  equal,  there  is  only  one  real  solution  between  0  and  TT. 

The  conditions  that  (48)  shall  have  a  double  root  are 

j  sin4  <p  =  M  sin  (<p  +  ra), 

(56) 

I  4  sin3  <p  cos  <p  =  M  cos  (<p  -f  m). 

The  solution  of  the  quotient  of  these  equations  for  tan  <p  is 


(57)  tan 


-  3  ±  V9  -  16  tan2  m 


2  tan  m 
It  follows  at  once  that  m  is  subject  to  the  condition 

9-16  tan2  m  ^  0 

in  order  that  the  double  root  shall  be  real.     Hence 
(58)         323°  8'  5  m  g  360°,         0  ^  m  ^  36°  52', 


220  LAPLACIAN   METHOD   OF   DETERMINING   ORBITS.  [123 

the  first  range  for  m  belonging  to  the  first  case,  represented  in 
Fig.  33,  and  the  second  to  the  second  case,  represented  in  Fig.  34. 

For  each  m  there  are  two  values  of  <p  defined  by  (57)  between 
0  and  TT.  In  the  first  case,  in  which  tan  m  is  negative,  tan  <p  is 
positive  whether  the  upper  or  the  lower  sign  is  used  before  the 
radical,  and  it  is  smallest  when  the  upper  sign  is  used.  Therefore 
the  value  of  <p  defined  by  (57)  when  the  upper  sign  is  used  is  that 
one  for  which  <pi  =  <pz  in  Fig.  33,  and  the  one  determined  when 
the  lower  sign  is  used  is  that  one  for  which  <pz  =  <p*.  When  m  has 
the  limiting  value  for  which  the  radical  vanishes  tpi  =  <p%  =  <p3. 
The  discussion  is  analogous  in  the  second  case  in  which  tan  m 
is  positive. 

The  limiting  values  of  <p,  defined  by  (57),  which  correspond  to 
the  limiting  values  of  m  as  given  in  (58),  are  respectively 

(59)  <p  =  116°  34',         <p  =  63°  26', 

and  for  both  of  these  values  of  <p  the  value  of  M  defined  by  (56) 
is  M  =  1.431.  This  is  the  maximum  M  for  which  (48)  can  have 
three  real  roots  between  0  and  IT.  In  order  that  the  three  roots 
shall  be  real  for  this  M  the  value  of  m  must  be  36°  52'  or  323°  8', 
and  the  three  roots  are  then  equal. 

Consider  the  first  case  and  suppose  m  starts  from  323°  8'  and 
increases  to  360°.  The  two  values  of  <p  defined  by  (57)  start 
from  63°  26'.  One  goes  to  0  and  the  other  to  90°.  The  two 
corresponding  values  of  M  start  from  1.431,  and  one  goes  to  0 
and  the  other  to  unity.  For  each  value  of  m  between  the  limits 
(58)  there  are  two  limits  between  which  M  must  lie  in  order  that 
(48)  shall  have  three  real  solutions.  In  constructing  a  table  of 
the  solutions  of  (48)  depending  on  the  two  independent  parameters, 
M  and  m,  these  limits  should  be  observed  in  order  to  reduce  the 
work  as  much  as  possible. 

123.  Differential  Corrections.  Suppose  the  approximate  solu- 
tion of  (48)  has  been  found  from  the  graphs  of  y\  and  yz,  or  by 
numerical  trials,  or  from  the  tables  of  the  roots  of  this  equation. 
Let  (po  represent  the  approximate  solution  and  <p0  +  A<p  the  exact 
solution.  The  problem  is  to  find  A<p. 

Let 

(60)  sin4  <PQ  —  M  sin  (<p0  +  m)  =  rj, 

where  rj  will  be  a  small  quantity  if  <p0  is  an  approximate  solution 
of  (48).  If  po  +  A<p  is  substituted  in  (48)  in  place  of  <p,  the  result 
expanded  as  a  power  series  in  A<p  becomes 


123]  DIFFERENTIAL    CORRECTIONS.  221 

—  77  =  [4  sin3  <po  cos  <p0  —  M  cos  (<pQ  +  ra)]A<p  +  [  ]  (A<p)2  +  •  •  • . 

This  power  series  can  be  inverted,  giving  A<p  as  a  power  series  in  rj. 
The  result  is 

/ai\       A                                               ~~  "n  I    n    2  _i_ 

(bl )       A<0   =  -3 — r— ; T-= 7 ; r  +  I   I  7?    H~ 

4  sin3  <PQ  cos  <po  —  M  cos  (<po  +  w) 

The  only  exception  is  when  the  coefficient  of  A<p  in  the  power 
series  in  A<p  is  zero.  This  is  the  second  of  equations  (56),  the 
conditions  for  a  double  root.  In  this  case  the  expression  for  A<p 
proceeds  in  powers  of  =•=  Vry.  In  practice  difficulty  arises  if  the 
coefficient  of  A<p  is  small  without  being  zero,  for  then  <p0  must  be 
very  close  to  the  true  value  of  <p  before  the  method  of  differential 
corrections  can  be  applied. 

The  higher  terms  of  (61)  can  be  computed  without  any  difficulty, 
but  they  rapidly  become  more  complex.  It  is  simpler  in  practice 
to  neglect  them  and  to  repeat  the  process  with  successive  improved 
values  of  (pe- 
lt is  possible  to  develop  a  more  convenient  method  for  com- 
puting the  differential  corrections  by  making  use  of  the  fact  that 
the  work  is  done  with  logarithms.  After  m  and  M  have  been 
computed  from  the  observational  data  the  approximate  solution 
of  (48)  can  be  determined  from  the  diagram.  The  curve  yi  can 
be  drawn  accurately  once  for  all.  The  better  known  sine  curve, 
in  this  case  flattened  or  stretched  vertically  by  the  factor  M,  can 
be  drawn  free  hand  with  sufficient  accuracy  to  enable  one  to  get  a 
very  approximate  estimate  of  the  value  of  <p.  Let  it  be  ^o.  The 
logarithms  of  the  right  and  left  members  of  (48)  will  be  computed 
and  they  will  of  course  be  found  to  be  unequal.  Let 
4  log  sin  <p0  —  log  M  —  log  sin  (<p0  -\-  m)  =  e. 

In  the  successive  approximations  only  the  first  and  third  of  these 
logarithms  will  be  changed.  The  tables  give  the  logarithms  of 
the  trigonometric  functions.  Let  the  tabular  difference  for  the 
logarithm  of  sin  <p0  and  sin  (<p0  +  5<p)  be  ci,  where  d<p  is  some 
convenient  increment  to  <po,  and  let  e2  be  the  corresponding  tab- 
ular difference  for  sin  (<p0  +  ni).  These  quantities  are  taken  down 
from  the  margins  of  the  tables  when  the  logarithms  of  sin  <p0  and 
sin  (<PQ  +  m)  are  taken  out.  Then  the  correction  A^?  is  given  by 
the  equation 


where  the  result  is  expressed  in  the  units  used  for  d<p.     This 


222 


LAPLACIAN   METHOD    OF   DETERMINING    ORBITS. 


[123 


method  is  so  convenient  in  practice  that  a  very  few  minutes  suf- 
fices in  any  case  to  find  the  solution  of  (48)  with  all  the  accuracy 
which  may  be  desired.  In  the  first  approximation,  where  the 
error  is  in  general  large,  one  degree  could  be  taken  for  d<p.  In  the 
later  approximations  10"  is  a  convenient  increment  because  the 
tabular  differences  of  the  logarithms  for  differences  of  10"  are 
given  on  the  margins  of  the  tables.* 

124.  Discussion  of  the  Determinant  D.  The  determinant  D, 
equation  (45),  enters  into  the  determination  of  the  constants  M 
and  m,  and  the  solution  becomes  indeterminate  in  form  if  it  is  zero. 
Consequently  it  is  important  to  find  under  what  circumstances 
it  vanishes. 

Suppose  the  determination  of  the  orbit  is  being  based  on  only 
three  observations.  Then  the  values  of  X,  X',  and  X",  which  occur 
in  D,  are  given  by  (31)  and  (32).  There  are  corresponding 
expressions  for  n,  /,  M";  v,  v'  ,  ""•  After  they  are  substituted  in 
(45)  the  determinant  D  can  be  factored  into  the  product  of  two 
determinants.  In  order  to  simplify  the  notation  let 


Pl  = 


,      , 


—  r2)  (TI  —  T3)  ' 


—  TI)  (r2  —  r3)  ' 


p    = 


—  TI)(T  —  TZ) 


—  TI)  (TS  —  TZ)  ' 

and  denote  the  derivatives  of  these  functions  with  respect  to  r 
by  accents.    Then 

"  D  =  AiAi 


(64) 


PI, 

Pi, 

p'; 

Ai  = 

PI, 

p;, 

pj 

p., 

PL 

pi 

Xi, 

xfc 

X3 

A2  = 

Ml, 

M2, 

M3 

. 

''i. 

''2, 

V3 

Consequently  D  can  vanish  only  if  A  i  or  A2  is  zero. 

*The  solution  of  (48)  depends  on  the  two  parameters  M  and  m',  if  there  were 
but  one  the  relations  between  it  and  <p  could  easily  be  tabulated.  In  spite  of 
the  two  parameters  Leuschner  has  extended  a  table  originally  due  to  Oppolzer 
from  which  the  solution  can  be  read  directly  with  considerable  approximation. 
It  is  table  xvi.  in  the  third  (Buchholz)  edition  of  Klinkerfues'  Theoretische 
Astronomie. 


124] 


DISCUSSION   OF  THE   DETERMINANT  D. 


223 


It  will  be  shown  first  that  AI  is  a  constant  which  is  distinct  from 
zero.  Since  it  is  formally  of  the  third  degree  in  T,  necessary  and 
sufficient  conditions  that  it  shall  be  independent  of  r  are  that 
A/  =  0  for  all  values  of  r.  The  derivative  of  a  determinant  is 
the  sum  of  the  determinants  which  are  obtained  by  replacing  suc- 
cessively the  columns  of  the  original  determinant  by  their  deriva- 
tives. Hence  A/  is  a  sum  of  three  determinants.  Since  the 
derivative  of  the  first  column  is  identical  with  the  second  column, 
the  first  of  these  determinants  is  zero  for  all  values  of  r.  Since 
the  derivative  of  the  second  column  is  identical  with  the  third, 
the  second  determinant  is  zero.  The  derivative  of  the  third 
column  is  zero,  and  therefore  the  third  determinant  is  zero. 
Hence  A/  is  identically  zero  and  AI  is  a  constant.  Its  value, 
which  is  easily  found  for  r  =  0,  is 


(65) 


^1  — 


T2T3,        T2  - 

hT3,        1 

2 

T,Ti,        T3J 

h-Ti,     1 

TiT2,        Tl   - 

hT2,        1 

—   T2)2(T2   —   T3)2(T3  —  Tl)2 

2 


(TZ  —  ri)(r3  —  T2)(r3  —  TI)  " 

This  determinant  is  distinct  from  zero  and  independent  of  the 
choice  of  the  epoch  tQ. 

In  order  to  interpret  A2  multiply  the  first,  second,  and  third 
columns  by  pi,  p2,  and  p3  respectively.  Then,  in  the  notation  of 
equations  (6),  the  determinant  A2  becomes 

,     a     £3 


The  right  member  of  this  equation  is  numerically  the  expres- 
sion for  six  times  the  volume  of  the  tetrahedron  formed  by  the 
earth  and  the  three  positions  of  C  with  respect  to  E.  The  volume 
of  this  tetrahedron  is  zero  only  if  the  three  positions  of  C  lie  in  a 
plane  passing  through  the  fourth  point  E.  This,  of  course,  is 
referring  the  position  of  C  to  E  as  an  origin.  A  simpler  way  of 
expressing  the  same  result  is,  the  determinant  A2  (and  therefore  D) 
is  zero  only  if  the  three  apparent  positions  of  C  as  observed  from  E 
lie  on  an  arc  of  a  great  circle. 

It  follows  from  (44)  that  if  D  is  zero,  DI  and  D2  are  also  zero 


224 


LAPLACIAN   METHOD    OF   DETERMINING   ORBITS. 


[125 


unless  R  =  r.  In  general,  the  expressions  for  p  and  p'  become 
indeterminate  when  D  is  zero,  and  they  are  poorly  determined 
when  D  is  small.  One  case  in  which  A2  and  D  are  always  zero  is 
that  in  which  C  moves  in  the  plane  of  the  earth's  orbit.  But  in 
this  case  there  are  only  four  elements  to  be  determined,  and  since 
each  observation  gives  a  single  coordinate  (the  longitude)  four 
observations  are  required. 

An  expression  for  A2  can  be  obtained  by  means  of  equations  (6). 
After  some  simple  reductions  it  is  found  that 

A2  =  cos  5i  cos  62  cos  63  [sin  (0:2  —  ai)  tan  £3 

/  (*£*\ 

+  sin  (0:3  —  at)  tan  61  +  sin  (ai  —  «3)  tan  62]. 

125.  Reduction  of  the  Determinants  DI  and  D2.  The  expres- 
sions for  DI  and  D2,  equations  (45),  become  as  a  consequence  of 
equations  (31)  and  (32)  and  corresponding  expressions  for  /*,  //,  v, 
and  / 


D1=   - 


Z),    =    + 


+  P2X2  +  P3X3, 
+  P2M2  +  PSMS, 

P\V\    +    P2^2    +    Ps^S, 
PiXi  rj"  P2X2  ~~h  P3X3, 

+  P2M2  +  P3M3> 
+   P2I'2   + 


P/-V  _|_     p    /\  _|_     p    /\  V" 

i  AI  ~(-  *s  A2  ~i~  i  3  A3,  -A 

Pl'/il  +  P2M2  +  Ps'/iB,  I" 

PiVi  +  P2V2  +  P3V3,  Z 

P//-V  I       p    //\  I       p    //\  V 

1    AI  ~t~  -t   2    A2  "p  ±   3    A3,  -A 

P//  |      p  //  i      p   //  ^ 

i   v\  H-  r^2   ^2  T  i  3   PS,  Z 


If  the  first  column  of  DI  is  multiplied  by  •—-  and  subtracted 
from  the  second  column,  the  result  is 

PA,     (P/P3  ~  PiP3')Xi  +  (P2T3  -  P2P3r)X2,     X 
(Pi'P,  -  PiPsOMi  +  (P.'Ps  -  P2P3OM2,      Y 


P3 


where 


Pv, 


-  PlP3')"l  +  (P2rP3   - 

x  =  PiXi  +  P2X2  +  P3X3, 


Pv- 

This  determinant  is  the  sum  of  the  two  determinants 

+    P-V  /P/P  PP/>\\_1_/'P/P  P 

-t   2A2,        \JL    i  J    3   —  i    IJL   3  ^Ai  "I"   \-t   2  *  8          *  ! 

+  P2M2,      (Pi'P,  -  PiPs')/*!  +  (P2rP3  -  P 


Pivi  +  P2^2,     (Pi'P,  - 


+ 


126]  REDUCTION    OF   THE   DETERMINANTS  £>i  AND  D2. 

and 


225 


Xs,     (Pi'Pj  -  PiPsOXi  +  (P2'P3  -  P2P3')X2,     X 
Ms,      (Pi'Pg  -  PiP3')Mi  +  (P2'P3  -  P2P3')M2,      Y 

( ~D    I  ~D  ~D    ~D   F\  I       /  ~D    I D  ~D    D    /\  ^7 

Vs,       \i  i  r 3  —  r \r 3  )v\  -f-  ^.r2  /^s  —  -r  2-t  3  j^2,      Z 

The  terms  in  X2,  M2,  and  vz  can  be  eliminated  in  a  similar  manner 
from  the  second  column  of  the  first  of  these  determinants.  Then 
each  of  the  determinants  is  the  sum  of  two  others,  and  the  reduced 
expression  for  DI  becomes 

Xi,     X2,     X 


=  -  (PfJ  -  P/P2) 


-  (P2P3'  -  P2'P,) 


-  (PsP/  -  P.'PO 


Mi,  M2,  Y 

vi,  v2,  Z 

\2)  A3,  J\. 

M2,  Ms,  Y 

\  \  y 

A3,  Al,  ^\. 

Ms,  Mi,  Y 


The  coefficients  of  these  determinants  are  needed  for  T  =  0.     It 
is  found  from  (63)  that 


1  f  2 


(r2  — 


P2P/   -   P,'P,    = 


P3P/  -  Ps'Px  = 


+ 


(r2  —  TI)(TS  —  r2)(r3  —  TI)  J 


(r2  —  ri)(r3  —  r2)(r3  —  TI)  ' 
Then  the  expression  for  DI  reduces  to 


(67) 


Xi,     X2,     X 

Vl,         J>2,         Z 

\3,        Xl,        X 

Ms,     MI,     Y 
PS,      v\)      Z 

P  =  (r2  —  TI)(TS  —  T2)(T3  —  TI). 


X2,     X3,    X 

M2,      Ms,      Y 

V*,        J/3,        Z 


16 


226 


LAPLACIAN   METHOD   OF   DETERMINING   ORBITS. 


[126 


In  a  similar  manner  the  expression  for  D2  reduces  to 


2r3 


(68) 


Xi,     X2,     X 
Mi,      M2,      Y 


Vz, 


2T2 


Ma,      Ms, 

P2,         *% 


X 


M3,       Ml; 


Each  of  the  determinants  in  the  expressions  for  DI  and  Da  can 
be  developed  in  a  form  similar  to  (66) . 

126.  Correction  for  the  Time  Aberration.  Since  the  velocity 
of  light  is  finite,  the  body  C  at  any  instant  is  apparently  where  it 
was  at  some  preceding  instant.  This  introduces  a  slight  error  in 
the  data  which  must  be  corrected,  if  accurate  results  are  desired, 
after  the  approximate  distances  have  been  determined.  Since  the 
velocity  of  light  is  very  great  and  the  apparent  motions  of  the 
heavenly  bodies  are  in  general  slow,  it  will  not  be  necessary  to 
know  the  distance  of  C  with  a  high  degree  of  accuracy  in  order  to 
correct  for  the  finite  velocity  of  light. 

Let  EI,  Ez,  and  E3  be  the  positions  of  the  observer  at  ti,  Z2, 
and  £3  respectively.  Let  the  observed  directions  of  C  at  these 


Fig.  36. 

epochs  be  EiCi,  E2Cz,  and  ESC3.  In  the  time  required  for  the 
light  to  go  from  C  to  E  the  former  will  have  moved  forward  in  its 
orbit  to  the  positions  pi,  pz,  and  p3,  which  are  its  true  places  at 
the  epochs  t\t  tz,  and  t$.  If  the  distances  are  known  the  observed 


127]  DEVELOPMENT  OF  X 


227 


coordinates  can  easily  be  corrected  for  these  slight  motions;  but 
this  changes  all  the  observed  data  of  the  problem  and  makes  it 
necessary  to  recompute  all  the  determinants. 

A  second  method,  which  is  more  convenient  in  practice,  is  to 
correct  the  times  of  the  observations.  The  body  C  passed  through 
the  points  Ci,  C2,  and  C3,  not  at  t1}  tz,  and  t3,  but  at  these  epochs 
diminished  by  the  time  required  for  light  to  move  from  Ci,  C2, 
and  C3  to  EI,  E2,  and  E3  respectively.  In  order  to  make  these 
corrections  to  the  epochs  it  is  necessary  to  know  EiCi  =  PI, 
EzCz  =  p2,  E3Cz  =  p3.  It  will  be  supposed  that  (48),  (46),  and 
(44)  have  been  solved  and  that  p  and  p'  are  known.  Then  the 
values  of  pi,  p2,  and  p3  are  given  with  sufficient  approximations  for 
present  purposes  by 

(Pi  =  P  +  P'TI, 
P2  =  P  +  P'T2, 
Ps  =  P  +  P'TS. 

Let  V  represent  the  velocity  of  light.  Then  the  epochs  at 
which  C  was  at  Ci,  C2,  and  C3  are 


(70) 


A  Pi  (p  -f-  P'TI) 

Tl       —       ATI        =        Tl       —       y      =        Tl       -  —y , 


P2 


—  Ar2  =  r2  —       =  r2  - 


(p  +  P' 


A  P3  (p  +  P' 

T3  —  Ar3  =  T3  —       =  r3  - 


Now  consider  the  correction  to  D,  DI,  and  D2.  In  D  only  the 
factor  AI  is  altered.  But  in  the  applications  only  the  ratios  of 
D  to  DI  and  D2  are  used,  and  the  latter  contain  AI  as  a  factor. 
Therefore  the  only  change  required  is  to  replace  TI,  r2,  and  r3 
by  TI  —  ATI,  T2  —  AT2,  and  T3  —  ATS  respectively  in  the  numerators 
of  the  coefficients  of  the  determinants  in  (67)  and  (68) . 

127.  Development  of  z,  y,  and  z  in  Series.  In  order  to  deter- 
mine the  corrections  which  should  be  added  to  X'  and  X",  so  as 
to  determine  the  elements  of  the  orbit  with  greater  accuracy,  it  is 
necessary  to  have  x,  y,  and  z  developed  as  power  series  in  T.  These 
quantities  satisfy  the  differential  equations 


228 


(71) 


LAPLACIAN   METHOD    OF   DETERMINING    ORBITS. 
ZX  X 


d?z  z 

-=--=-uz. 


[127 


It  is  shown  in  the  theory  of  differential  equations  that  the  solu- 
tions of  differential  equations  of  this  type  are  expansible  as  power 
series  of  the  form 


X  =  X0       XQT  r  XQT 

y  =y«  +  y«'r  +  ^0'V  +  ^0"V 

So  =  20  +  Zc'r  +  ^o'V2  +  W'r3   +  ^oivr4  +  yin  ZoV  + 
It  is  found  from  (71)  and  its  successive  derivatives  that 


(72) 


The  coefficients  of  the  series  for  y  and  z  differ  only  in  that  yQ,  yQ' 
and  z0,  z</  appear  in  place  of  XQ,  XQ  respectively.     Therefore 

x  =  fxQ  +  gxo, 
y  =fy<> 


(73) 


g  =  r  —  JwoT3  —  TV^o'f4  —  Ts-5-(3l£o"  —  UQ2)r5  +  •  •  • . 
In  order  to  have/  and  g  in  a  form  for  practical  use  the  derivatives 
of  u  must  be  expressed  in  terms  of  XQ,  yo,  ZQ,  XQ',  y<>',  and  z0r.     La- 
grange  has  done  this  very  elegantly  by  introducing  p  and  q  by  the 
equations 


(74) 


, 
P  =  2  fc  = 


Then  it  is  found  that 


128] 


THE   HIGHER   DERIVATIVES    OF   X,  fJL}  v. 


229 


3  dr 

I3- 

r4  dr 


3   1  dr2 

i  ?T  ~^~ 

r4  2r  dr 


P 


By  means  of  these  equations  and  their  successive  derivatives  the 
coefficients  in  the  series  for  /  and  g  can  be  expressed  as  polynomials 
in  u,  p,  and  q.  The  expressions  for  /  and  g  become 


(75) 


The  derivatives  of  x,  y,  and  z  can  be  determined  from  equations 
(73)  and  (75).     For  example 

"  =  f'"x0  +  </"V, 
(76)  -j  ziv  =  f^xo  +  g^xo', 


128.  Computation  of  the  Higher  Derivatives  of  X,  M>  v.    The 

values  of  X,  X',  and  X"  determined  by  equations  (31)  and  (32)  are 
only  approximate  because  c3,  c4,   •  •  •  were  unknown.     But  after 
the  higher  derivatives  become  known  these  coefficients  are  obtain- 
able, and  the  approximate  values  can  be  corrected. 
The  third  derivatives  of  equations  (8)  are 

"X  +  3p"X'  +  3p'X"  +  PX'"  =  x'"  +  X"', 
(77)       J  p"'M  +  3p'V  +  3pV  +  PM'"  =  y"r  +  Y"f, 


3p'V  +  3P 


The  left  members  of  these  equations  involve  the  four  unknowns 
p'",  X'",  IJL"',  and  /",  the  first  and  second  derivatives  having 
been  determined  approximately  by  equations  (31),  (32),  and  (44); 
but  the  unknowns  are  not  independent  because  X,  n,  v,  and  their 
derivatives  satisfy  the  relations 


XXr  +  MM'  +  w'  =  0, 

XX"  +  MM"  +  vv"  +  X'2  +  M'2  +  v*  =  0, 

XX'"  +  MM"'  +  vv'"  +  3(XrX"  +  MM"  +  v'v")  =  0. 


230  LAPLACTAN   METHOD   OF   DETERMINING    ORBITS.  [129 

Consequently  if  equations  (77)  are  multiplied  by  X,  n,  and  v 
respectively  and  added,  the  result  is 

p'"  =  3P'(X/2  +  M'2  +  "'2)  +  3p(X'X"  +  MV'  +  v'v") 

+  (*'"  +  X'")\  +  (y"f  +  F'")M  +  (*'"  +  Z">, 

which  uniquely  defines  prrr.  Then  X'",  /*'",  and  /"  are  deter- 
mined by  (77)  because  x"f,  y"f,  z"'  are  given  by  (76)  and  X'", 
Y'",  and  Z"'  can  be  found  from  the  Ephemeris. 

The  quantities  Xiv,  juiv,  and  viv  can  be  computed  in  a  similar 
way  by  taking  the  derivatives  of  (77)  and  reducing  by  means  of 
the  relations  among  X,  n,  and  v. 

129.  Improvement  of  the  Values  of  x,  y,  2,  x',  yf,  z'.  After 
D,  Z)i,  and  D2  have  been  found  from  (65),  (66),  (67),  and  (68) 
equation  (48)  can  be  solved,  and  then  x,  y,  z  and  their  first  deriv- 
atives can  be  determined  from  (8)  and  their  first  derivatives. 
These  results  are  only  approximate  because  of  the  errors  to 
which  X,  fi,  v,  X',  IJL',  and  v'  are  subject,  and  the  problem  is  to 
correct  them  after  X'",  //'",  •  •  •  have  been  determined. 

It  follows  from  the  first  equation  of  (28)  that 

C3    =  JX'",        c4  =  AXiv,         ......  . 

Then  equations  (35)  give 

ACo    =     —    iX///TlT2T3    —   -2J¥XivTiT2T3(Tl   +   T2  +   T3)    +    '  '  '  , 

ACi  =  +  iX'"(riT2  +  T2T3  +  ran) 

+  1&VV(T1   +   T2)(T2  +   T3)(T3  +   Ti)    +    '  '  •, 

AC2  =  -  JX"'(TI  +  r2  +  TJ) 

-   T^W  +  T22  +  T32  +  TlT2  +  T2T3  +   T,Ti)   +    •  •  •  , 

and  the  expression  for  X  becomes 

X  =  C0  +  Ac0  +  (ci  +  Aci)r  +  (c2  +  Ac2)r2 


where  Co,  Ci,  and  c2  are  the  approximate  values  of  the  coefficients 
of  the  series  which  are  obtained  from  (31)  and  (32)  by  putting 
r  equal  to  zero.  There  are  corresponding  equations  for  /z  and  v. 
With  these  more  nearly  correct  values  of  X,  X',  X",  •  •  •  ,  the  de- 
terminants D,  D],  and  Z>2  are  computed  from  (45),  <p  is  determined 
from  (48),  p  and  p'  from  (44),  and  x,  y,  z,  x',  y',  z'  from  (8)  and 
their  first  derivatives.  Then  still  higher  derivatives  of  X,  /*,  v  can 


130]  THE   MODIFICATIONS    OF   HARZER   AND    LEUSCHNER.  231 

be  computed  and  still  more  nearly  exact  values  of  X,  X',  and  X" 
determined,  or  the  elements  can  be  determined  from  x,  y,  z,  x', 
y'j  z'  by  the  methods  of  chap.  v. 

There  are  two  principal  objections  to  the  method  of  Laplace. 
One  is  that  it  is  necessary  to  recompute  all  determinants  and 
auxiliaries  at  each  stage  of  the  approximation,  each  of  which 
costs  a  very  considerable  amount  of  labor.  The  other  is  that 
the  method  depends  upon  the  motion  of  the  observer  through  the 
equations  by  means  of  which  X" ',  Y",  and  Z"  were  eliminated 
from  (9) .  Obviously  all  that  is  really  fundamental  in  the  problem 
is  that  C  shall  have  been  observed  from  definite  known  places 
and  that  it  shall  move  about  the  sun  in  accordance  with  the  law 
of  gravitation. 

130.  The    Modifications    of    Harzer    and    Leuschner.    The 

method  of  Laplace  for  determining  orbits  has  not  been  found 
very  satisfactory  in  practice.  The  reason  seems  to  be  that  the 
conditions  that  the  first  and  third  observations  shall  be  exactly 
satisfied  are  not  directly  imposed  as  they  are,  for  example,  in  the 
method  of  Gauss.  To  remedy  this  defect  Harzer  proposed*  the 
plan  of  so  determining  x,  y,  z,  x',  y',  z'  by  differential  corrections, 
after  their  approximate  values  have  been  found,  that  the  three 
observations  shall  be  exactly  fulfilled.  If  more  than  three  obser- 
vations are  under  consideration,  they  cannot  in  general  be  exactly 
satisfied,  and  the  adjustments  are  then  made  by  the  method  of 
least  squares. 

It  will  be  sufficient  here  to  sketch  the  method  of  making  the  dif- 
ferential corrections.  The  right  ascensions  and  declinations  are 
expressed  in  terms  of  the  coordinates  and  components  of  velocity 
at  tQ  by 

pX  =  fx0  +  gxo'  +  X, 

PM  =  fyo  +  gyo'  +  Y, 

.pv  =  fz0  +  gz0'  +  Z, 

which  are  obtained  by  substituting  equations  (73)  in  equations  (8). 
The  right  ascension  and  declination  enter  through  X,  /*,  and  v  of 
equations  (6).  The  result  can  be  indicated 

|  a  =  F(x0,  ?/o,  Zo,  XQ'J  y0',  z</), 

I  5  =  G(xQ,  2/0,  Zo,  XQ',  2/0',  z0'). 

*  Astronomische  Nachrichten,  Nos.  3371-2  (1896). 


232  GAUSSIAN   METHOD    OF   DETERMINING   ORBITS.  [131* 

From  these  equations  the  variations  in  a  and  5,  which  are  the 
known  differences  between  the  observations  and  the  approximate 
theory,  are  expressed  in  terms  of  the  variations  in  x0,  •  -  - ,  z0',  which 
are  required.  The  relations  are 

dF  .       .  dF  .       .  dF .          dF  .        ,    dF  .        .   dF  . 


dG  .       .  dG  .       .  dG  .       .    dG  .     ,  ,    dG  .     ,  .    dG  .    , 

^  A*° + W«Ayo + ^A0° +  to?  AXQ  +  w*»  +  s?^°  • 

In  forming  the  partial  derivatives  it  must  be  remembered  that 
XQ,  •  •  • ,  2o'  enter  through  /  and  gf  as  well  as  explicitly.  When  these 
equations  are  written  for  three  dates  they  become  equal  to  the 
number  of  arbitraries,  viz.,  A#0,  •  •  • ,  A20',  and  consequently  deter- 
mine them  uniquely  provided  the  determinant  of  their  coefficients 
is  distinct  from  zero.  The  circumstances  under  which  it  vanishes 
have  not  been  investigated.  If  there  are  more  than  three  obser- 
vations the  number  of  equations  exceeds  the  number  of  arbitraries 
and  the  method  of  least  squares  is  employed. 

When  the  date  of  the  second  observation  is  taken  as  the  origin 
of  time  and  the  number  of  observations  is  only  three,  the  number 
of  equations  of  condition  reduces  to  four  which  in  general  can  be 
satisfied  by  suitably  determining  Ap0,  Azo',  AT/O',  and  Az0'.  This 
is  the  procedure  adopted  by  Leuschner*  to  abbreviate  the  method 
of  Harzer.  In  its  simplified  form  the  method  has  been  found  very 
convenient  in  practice  and  has  led  to  highly  satisfactory  results. 

II.     THE  GAUSSIAN  METHOD  OF  DETERMINING  ORBITS. 
131.  The  Equation  for  p2.     Equations  (19)  are  fundamental  in 
the  method  of  Gauss.     If  the  geocentric  coordinates  are  intro- 
duced by  equations  (8),  equations  (19)  become 

[2,  3]piXi  -  [1,  3]P2X2  +  [1,  2]p3X3 

=  [2,  3]Zi  -  [1,  3]X2  +  [1,  2]X,, 

[2,  3]plMl  -  [1,  3]p2M2  +  [1,  2JP3M3 

=  [2,3]ri-[l,  3]F2  +  [1,2]F3, 
[2,  3]pin  -  [1,  3W2  +  [1,  2]P8», 

=  [2,  3]Z,  -  [1,  3]Z2  +  [1,  2]Z8. 

The  left  members  of  these  equations  are  linear  in  the  three  un- 
knowns pi,  p2,  and  p3.  Their  solution  for  p2  is 

*  Publications  of  the  Lick  Observatory,  vol.  vn.,  Part  1  (1902). 


(80) 


131] 


THE   EQUATION   FOR   p2. 


233 


(81)  - 


D 


Mb      M2,      M3 

Vl,         *2t         V* 


?     X3 


Mi,     [2,  3]Ft-[l,  3]F2+[1,  2]F3, 


=  [2,  3][1,  2] 

-[1,  3]Z2+[1,  2]Z8, 
The  determinant  D  is  the  sum  of  three  determinants 
D  = 


(82)  < 


Mb 


Ma 


a,     Xs 


M3 


7)  (2)    = 


Mb 
"b 


Consequently  the  first  equation  of  (81)  becomes 
(83)  A2P2  =  - 


Suppose  tz  is  taken  as  the  origin  of  time.     Then  it  follows  from 
equations  (73)  that 


The  expressions  for  the  ratios  of  triangles  then  become 

[2,  3]  _ 
[1,  3]  " 
[1,2] 
[1,  3]  ~ 


(84) 


The  numerators  and  denominators  of  the  expressions  for  the  right 
members  of  these  equations  are  found  from  (75)  to  be  expansible 
as  power  series  in  TI  and  r3.  But  in  order  to  simplify  (83)  it 
is  convenient  to  let 


234 


GAUSSIAN   METHOD    OF  DETERMINING    ORBITS. 


[131 


(85) 


k(ts  —  ti)  =  T3  —  TI  =  2r, 


Ck(tt  -  ti) 

[TI  =   -  r 


+  e,          r3  =  +  r  +  e, 


where  e  is  in  general  small  compared  to  r,  and  will  be  supposed  to 
be  of  the  order  of  r2.  Then  the  expressions  for  the  ratios  of  the 
triangles  become 

[2,3]  _        +ga        _1  1      ,         re 


(86)  ^ 


[1,  2]  _         -  flfi        =l__l   ,    1 
[1,3]     /i03-/30i     2     2r"t"4 


re 
12 


where  all  terms  up  to  the  sixth  order  have  been  written.     The 
quantity  u  is  defined  by  u  =  —  and  p  and  q  are  defined  in  (74). 
On  making  use  of  equations  (86) ,  equation  (83)  becomes 

r2 


A2P2    =    K 


PK,+ 


re 


QK,, 


where 


AI,     -^-i,     ^3            Aij     ./L  2,     AS 

1 

1 

*--2 

Mi,      ^i,      Ms    +     Mi,      YZt      M3     —  2 

Vi,        Zi}        Vs.                 Vi.        Z*>.        Vz 

X 

1,        »*1|        A3 

*y                   ••/ 

Xl,       ^L3,       Xs 

^1    =     — 

Ml,      Ylt      M3 

.— 

Mi,      Y3,      Ms 

t 

vi,     Zi,      */3 

Vl,         Z8,         *>3 

AI,     .A.  i,      AS 

^  1?        *^-  3j        *^3 

K.%  =  — 

Mi,      YI,      Ms 

+ 

Ml,      Y3,      M3 

. 

V 

Li        ^1,        "3 

vi,     Z8,     v, 

Xi, 


M3 


The  right  members  of  the  expressions  for  K, 
giving  the  simpler  expressions 


I,  and  K2  add, 


131] 


THE   EQUATION    FOR   p2 


235 


(88) 


K=~2 


Mi, 


+  ^3  -  2F2, 

+  Z3  -  2Z2, 


—   \i 

-  Ml 


+ 


Mi,  Fi  +  Y9,  Ms  -  Mi 

I/I,  Zi    +     ^3,  ?3   —    Vi 

Xi,  Xa  —  ^i,  Xs  —  Xi 

Mi,  ^3  —   Yi,  Ms  —  Mi 


Consider  equation  (87).  The  determinant  A2  by  which  the  left 
member  is  multiplied  is  given  in  terms  of  the  on  and  5;  by  (66), 
which  appeared  in  'the  method  of  Laplace.  It  can  also  be  written, 
by  properly  combining  columns,  in  the  form 


A2  = 


Xi,     X2,     Xs 


Mi,     M2,      Ms 


Xi, 


Mi, 


Xi  +  Xs  —  2X2, 
Mi  +  Ms  —  2/z2, 
v\  +  J>3  —  2i'2, 


Xs  — 


Ms  —  Mi 


If  Xf,  MI,  ^i  are  replaced  by  the  series  (28),  taking  r2  =  0,  the 
second  column  is  of  the  second  order  and  the  third  column  is  of 
the  first  order  in  the  time-intervals.  Therefore  A2  is  of  the  third 
order. 

Since  the  left  member  of  (87)  is  of  the  third  order  the  right 
member  also  must  be  of  the  third  order.  The  second  column  of 
the  expression  for  K,  the  first  equation  of  (88),  is  of  the  second 
order,  and  the  third  column  is  of  the  first  order.  Therefore  K  is 
of  the  third  order.  The  determinant  KI  is  of  the  first  order  and 
Kz  is  of  the  second  order.  The  former  is  multiplied  by  r2,  which 
is  of  the  second  order,  and  the  latter  by  re,  which  is  of  the  third 
order.  In  a  preliminary  determination  of  an  orbit  the  terms  of 
higher  order  may  be  omitted,  after  which  (87)  becomes 


=  K 


4r23 


This  equation  is  of  the  same  form  as  the  first  of  (44) ,  and  involves 
the  two  unknowns  p2  and  r2.  They  are  expressible  in  terms  of 
a  single  unknown  <p  by  means  of  equations  (46)  affected  with  the 


236 


GAUSSIAN   METHOD   OF  DETERMINING   ORBITS. 


[132 


subscript  2.     The  resulting  equation  has  exactly  the  same  form 
as  (48),  and  its  solution  gives  approximate  values  of  p2  and  r2. 

132.  The  Equations  for  pt  and  p3.  Equations  (80)  are  linear 
in  pi  and  p3,  and  these  quantities  can  be  determined  from  any 
two  of  the  three  equations.  The  two  to  be  used  in  practice  are 
those  for  which  the  determinant  of  the  coefficients  of  pi  and  p3  is 
the  greatest,  for  they  will  best  determine  these  quantities. 

The  solution  of  the  first  two  equations  of  (80)  for  pi  and  p2  if 
they  are  written  first  in  determinant  form,  and  if  they  are  then 
expanded  as  a  sum  of  determinants,  is 


Pi  = 


(89)  H 


'      Xt, 

X3 

Mi, 

M3 

Xi, 

X3 

Mi, 

M3 

P3    = 


Xi,     X3         [i 

,3] 

X2,     X3 

Ylt     M3         [2 

,3] 

F2,      Ms 

,    [1,  2] 

F3, 

X3 

Ms 

+   P2  [2 

,3]    X 

h  [2,  3] 

,3]    p 

2,  3]    Xi,     Xi 

[1,3]    Xi, 

X2 

U>21    MI,     Fi 

U,2]    Ml, 

F2 

Xi, 

x, 

[1 

,3]    X 

Mi, 

F3 

^^U^]      „ 

The  solution  of  the  first  and  third  equations  of  (80)  differs  from  this 
only  in  that  the  MI  are  replaced  by  the  vi}  and  the  F;  by  the  Zi\ 
and  the  solution  of  the  second  and  third  equations  of  (80)  can  be 
obtained  from  (89)  by  changing  the  Xt,  jj,i}  Xi,  and  F»  to  m,  Vi, 
Yi,  and  Zi  respectively. 

After  pi,  p2,  and  p3  have  been  computed  the  correction  of  the 
time  for  the  time-aberration  can  be  computed.  The  method  was 
explained  in  Art.  126. 

133.  Improvement  of  the  Solution.  The  results  so  far  obtained 
are  only  approximate  because  only  the  first  term  of  P  was  retained 
while  the  term  in  Q  was  entirely  neglected.  Having  found  an 
approximate  solution  it  is  easy  to  correct  it.  The  values  of  pi,  p2, 
and  p3  are  known,  and  the  corresponding  values  of  r  can  be  found 
at  each  of  the  three  epochs  from 

r2  =  p2  +  R2  -  2PR  cos  ^, 

which  expresses  the  fact  that  S,  E,  and  C  form  triangles  at  the 
dates  of  the  three  observations.  After  ri}  r2,  and  r3  have  been 


134] 


RATIOS  OF  TRIANGLES  BY  METHOD  OF  GAUSS. 


237 


found  the  first  and  second  derivatives  of  r  at  t  =  tz  can  be  found 
by  the  method  of  Art.  113.  Then  equations  (74)  define  p  and  q 
after  which  more  approximate  values  of  P  and  Q  can  be  determined. 

134.  The  Method  of  Gauss  for  Computing  the  Ratios  of  the 
Triangles.  Equation  (83),  which  is  fundamental  in  determining 
p2  and  r2,  involves  two  ratios  of  triangles.  It  follows  from  (86) 
that  they  can  be  written  in  the  form 


(90) 


[2,3]  1       _e_      Pi 

[1,3]  2~h2r~hr23' 

[1,2]  =1        e_      P* 

.[1,3]  '2      2r~t~r23* 


Consequently,  if  the  ratios  of  the  triangles  can  be  determined 
PI  and  P2  can  be  found  from  these  equations.  One  of  the  im- 
portant features  of  the  method  of  Gauss  is  a  convenient  means  of 
determining  the  ratios  of  the  triangles.  In  order  to  apply  this 
method  it  is  necessary  to  find  the  inclination  and  node  of  the  orbit 
and  the  argument  of  the  latitude  at  the  dates  of  the  observations. 
Since  the  geocentric  coordinates  are  all  known  after  pi,  p2,  p3 
have  been  determined,  the  heliocentric  coordinates  can  be  com- 
puted. Suppose  ecliptic  coordinates  are  used  and  that  the 


Fig.  37. 

longitudes  and  latitudes,  as  well  as  the  distances,  are  known 
at  ti,  tz,  and  £3.  The  inclination  is  less  or  greater  than  90°  according 
as  Z3  is  greater  or  less  than  li.  Then  it  follows  from  the  spherical 
triangles  Ci&li  and  C3&Z3  that 

{tan  i  sin  (li  —  &)  =  tan  61, 
tan  i  sin  (h  —  &)  =  tan  63. 

But  ?3  —  &  =  (h  —  li)  +  (h  —  &);  therefore  these  equations 
become 


238  GAUSSIAN   METHOD    OF   DETERMINING    ORBITS.  [135 


tan  i  sin  (li  —  Q>)  =  tan  6 


1, 


tan  i  cos  (Z;  -  ft)  =   tan  6.  -tan  6t  cos  (I.-  t.) 

sm  (£3  —  li) 

which  determine  i  and  &  uniquely  since  the  quadrant  of  i  is  al- 
ready known  from  the  sign  of  ls  —  l\. 

The  longitude  of  C  from  the  node  is  called  the  argument  of  the 
latitude.  It  follows  from  Fig.  37  that 

(cos  (lj-  —  &)  cos  bj  =  cos  Ujy          (j  =  1,  2,  3), 
sin  (7/  —  &)  cos  6,-  =  sin  u}-  cos  i, 
sin  6y  =  sin  Uj  sin  i, 

which  uniquely  define  HI,  u*,  and  us. 

Let  A  equal  the  area  of  the  sector  contained  between  the 
radii  r\  and  r2  and  the  orbit.  Then  the  ratio  of  the  area  of  the 
sector  to  the  area  of  the  triangle  contained  between  r\  and  r2  is 


(93)  = 


r2        ri  r2  sn  uz  —  Ui 

where  p  now  represents  the  parameter  of  the  conic.  Suppose  the 
corresponding  ratios  for  ts  —  ti  and  Z3  —  tz  have  been  found;  then 
the  ratios  of  the  triangles  are  known.  The  method  of  Gauss 
depends  upon  the  determination  of  these  ratios.  Each  of  these 
quantities  is  denned  by  two  simultaneous  equations  in  two  un- 
known quantities. 

135.  The  First  Equation  of  Gauss.     The  polar  equation  of  the 
conic  gives 

—  =  1  +  e  cos  0i, 


whence 
(94) 


=  1  +  e  cos  v2'f 

17*2 


p  —     — -  =  2  +  e(cos  vi  +  cos 


Since  v%  —  v\  =  uz  —  u\  is  known,  the  only  unknown  in  the  right 
member  of  this  equation  is  e  cos  (  -^—= — -  )  ,  which  will  now  be 
eliminated.  From  the  equations  of  Art.  98  it  follows  that 


135] 


THE   FIRST  EQUATION    OF   GAUSS. 


239 


(95) 


r—  i  l~77~  I 

Vri  cos  2*  =  Va(l  -  e)  cos  —, 

Vn  sin  ^  =  Va(l  +  e)  sin  y  , 

/  —        ^2        r~7^  -  \         **J 
Vr-2  cos  —  =  Va(l  —  e)  cos  -^  , 

Vr^  sin  ^  =  Va(l  +  e)  sin  —  2  . 


From  these  equations  it  is  found  that 

(E*- 
=  acos(  --  ^  --     ~  ae  cos 


ir2  cos 


=  a  cos 


~  ae  cos 


(F   -\-  F1  \  /        I    v  \ 

2     -  }  and  solving  for  e  cos  (    2        -  j  , 

it  is  found  that 
e  cos 


As  a  consequence  of  this  equation  (94)  reduces  to 


rt   i  -         fvz  —  vi\          /  E 
-  2Vr1r2cos(  -^  —  1  cos  ( 


On  eliminating  p  from  this  equation  and  (93)  the  equation 


(96) 


O/  --  /^2-^l\ 

|  ri  +  r2-2Vr1r2cos(  —  —  ••  )  cos 


is  obtained.     In  order  to  simplify  it  let 

t>2  —  vi  =  u<z  —  HI  =  2f, 
s-  E!  =  20, 


(97) 


m 


4  Vr 


rJ_        1 
f       2" 


cos 


240 


GAUSSIAN   METHOD    OF   DETERMINING    ORBITS. 


[136 


Then  the  expression  for  rj2  reduces  to 

m2 


in  which  rj  and  g  are  the  unknowns.     This  is  the  first  equation  in 
the  method  of  Gauss. 

136.  The  Second  Equation  of  Gauss.  An  independent  equation 
involving  77  and  g  will  now  be  derived.  It  will  be  made  to 
depend  upon  Kepler's  equation,  thus  insuring  its  independence 
of  (98)  which  was  derived  without  reference  to  Kepler's  equation. 
The  first  equations  are 


Ml  = 


=     2-e  sn 


whence 


*i)      o 

'•   =2g  -2e  sin  g  cos 


(ET       i      TfJ    \ 
-2—^  —  l  j  must  be  eliminated  in  order 

to  reduce  this  equation  to  the  required  type.     On  making  use  of 
the  first  equation  following  (95),  it  is  found  that 


(99) 


2g  -  sin  2g  +  2 


sin  g  cos  /. 


It  remains  to  eliminate  a.    By  Art.  98 


whence 


—  =  1  —  e  cos 


—  =  1  —  e  cos  Ez; 
a 


r-2 


=  2  —  2e  cos  Q  cos 

#2   + 


(rr      I     F   \ 
2—~ — -  )  by  the  first  equation  following 


(95)  this  equation  becomes 

1  2  sin2  g 


r2  — 


cosgr  cos/ 


137]  SOLUTION    OF   EQUATIONS    (98)   AND    (101).  241 

which  becomes  as  a  consequence  of  the  expression  for  if 


On  eliminating  a  between  (99)  and  (100),  it  is  found  that 

(101)  4_4=^_TL»n^.i 

m?        m2  sin3  g 

which   is   the   second   equation   in  77   and  g.     There  are  similar 
equations  for  the  time-intervals  ts  —  ti  and  ts  —  22. 

137.  Solution  of  (98)  and  (101).  It  follows  from  the  definition 
of  77  that  it  is  positive  if  the  heliocentric  motion  in  the  orbit  is 
less  than  180°  in  the  interval  tz  —  t\.  It  will  be  supposed  in  what 
follows  that  the  observations  are  so  close  together  that  this  con- 
dition is  fulfilled. 

Let 


(102) 


2g  -  sin  2g  _  v 

: — a —   -A. 

sin3  g 


Oh  eliminating  t\  from  (98)  and  (101)  and  making  use  of  (102), 
it  is  found  that 

(103)  m=-(l  +  a)*  +  X(l  +  x)*. 

The  quantity  X  must  now  be  expressed  in  terms  of  x,  after  which 
(103)  will  involve  this  quantity  alone  as  an  unknown.  This  will 
be  done  by  first  expressing  X  in  terms  of  gr,  and  then  g  in  terms  of  x. 
The  following  are  well-known  expansions  of  the  trigonometrical 
functions  : 

f  sin  2g  =  20  -  f<73  +  TV  -  " 


whence 

*  ~ 


nrvn 


From  the  first  of  (102)  it  follows  that 
g  =  2  sin-K**)  =  2x*  +  ^  + 


17 


242  GAUSSIAN   METHOD    OF   DETERMINING    ORBITS.  [137 

Then  (104)  becomes 

J  =  4[\    ,  6      ,  6-8   .   ,  -6-8-10 

or 

X  = 


Let 


6         6-8   „      6-8-  10 


3_JLF         JL   2  _     52   ^  , 
4      10  [X      35 x       1575^ 


If  \g  is  a  small  quantity  of  the  first  order,  x  is  of  the  second  order 
and  £  is  of  the  fourth  order. 
From  (98)  it  is  found  that 

(106)  x  =  ^  -  I 
Let 

(107)  ft  =  _"-_. 


then  (101)  may  be  written 

1  _m2X  _ 
*-  ^jT= 

from  which  it  is  found  that 

(108)  n3  ~  f  ~  hr,  -  g  =  0. 

If  £  were  known  h  would  be  determined  by  (107)  and  f\  by  (108), 
which  has  but  one  real  positive  root.  In  the  first  approximation 
compute  h  assuming  that  the  small  quantity  £  is  zero ;  then  find  the 
real  positive  root  of  (108).  Or,  instead  of  computing  the  root, 
use  may  be  made  of  the  tables  which  have  been  constructed  by 
Gauss,  giving  the  real  positive  values  of  T\  for  values  of  h  ranging 
from  0  to  0.6.  *  The  value  of  x  is  then  computed  by  (106)  and  the 
value  of  £  by  (105) .  f  With  this  value  of  £,  h,  and  rj  are  recomputed, 
and  the  process  is  repeated  until  the  desired  degree  of  precision 
is  attained.  Experience  has  shown  that  this  method  of  computing 

*This  table  is  XIII.  in  Watson's  Theoretical  Astronomy,  and  VIII.  in 
Oppolzer's  Bahnbestimmung . 

t  The  value  of  £  with  argument  x  is  given  in  Watson's  Theoretical  Astronomy, 
Table  XIV.,  and  in  Oppolzer's  Bahnbestimmung,  Table  IX. 


138]  DETERMINATION    OF   THE   ELEMENTS   a,  6,    AND   a>.  243 

the  ratio  of  the  sector  to  the  triangle  converges  very  rapidly,  even 
when  the  time-interval  is  considerable. 

The  species  of  conic  section  is  decided  at  this  point,  the  orbit 
being  an  ellipse,  parabola,  or  hyperbola  according  as  x  is  positive, 

zero,  or  negative;  for,  x  =  sin2  -|-  =  sin2-  (E2  —  EJ,  and  E2  and 

EI  are  real  in  ellipses,  zero  in  parabolas,  and  imaginary  in  hyper- 
bolas. 

Gauss  has  introduced  a  transformation  which  facilitates  the 
computation  of  I  which  was  denned  in  the  last  equation  of  (97)4 
Let 

fe  =  tan  (45°  +  «'),        0°  ^  a/  ^  45°; 
whence 

Tj-^*  =  ^  +  J^  =  tan2  (45°  +  co')  +  cot2  (45°  +  a/), 
or 

fli=^=  2  +  4tan22a/. 


Then  the  last  equation  of  (97)  becomes 

sin2    +  tan2  2o/ 


cos/ 

138.  Determination  of  the  Elements  a,  e,  and  o>.  After  g  has 
been  found  by  the  method  of  Art.  137  it  is  easy  to  obtain  the  ele- 
ments a,  e,  and  co.  The  major  semi-axis  a  is  defined  by  the  last 
equation  on  page  240,  or  by  the  preceding  equation  for  the  longer 
time-interval  23  —  h, 

(109)  a 


2  sin2  g 

The  parameter  of  the  orbit  p  is  determined  by  equation  (93)  . 
Since 

(110)  p  =  a(l  -  e2)     or     p  =  a(e*  -  1) 

according  as  the  orbit  is  an  ellipse  or  hyperbola,  e  is  determined 
when  a  and  p  are  known. 

If  the  angle  v  is  computed  from  the  perihelion  point  it  is  related 
to  the  heliocentric  distances  and  e  and  p  by  the  polar  equation  of 
the  conic, 

t  Theoria  Motus,  Art.  86. 


244  SECOND   METHOD   OF   DETERMINING   a,   6,   AND   CO.  [139 


Either  of  these  equations  determines  a  value  of  v  since  r  is  known 
at  fc,  tz,  and  £3,  and  then  co  is  determined  by 

(112)  co  =  Ui  -  v{. 

139.  Second  Method  of  Determining  a,  e,  and  to.  The  method 
of  Gauss  depends  upon  the  complicated  formulas  of  Arts.  135  and 
136.  If  the  higher  terms  of  P  and  Q,  equations  (86),  give  suf- 
ficiently accurate  values  of  the  ratios  of  the  triangles,  there  is 
another  method  *  which  is  simpler  and  especially  advantageous 
when  the  intervals  between  the  observations  are  not  very  great. 
The  data  which  will  be  used  in  the  solution  are  ri,  u\]  rz,  uz]  r3,  w3, 
the  heliocentric  coordinates  at  ti,  t»,  and  ts. 

The  elements  i  and  &  can  be  computed  by  equations  (91), 
which  are  valid  for  any  orbit.  The  difficulties  all  arise  in  finding 
a,  e,  co.  Let  the  parameter  p  be  adopted  as  an  element  in  place 
of  the  major  semi-axis  a.  It  is  more  convenient  in  that  it  does  not 
become  infinite  when  e  equals  unity,  and  it  is  involved  alone  in 
the  equation  of  areas, 

k  -Jpdt  =  r*dv  =  r*du. 
The  integral  of  this  equation  is 

(113)  kJp(t3  -  ti)  =  C*r*du. 


If  r2  were  expressed  in  terms  of  u  the  integral  in  the  right  member 
could  be  found,  when  the  value  of  p  would  be  given.  It  will  be 
shown  from  the  knowledge  of  the  value  of  r2  when  u  =  HI,  Uz,  u3) 
viz.,  r2  =  ri2,  r22,  r32,  that  r2  can  be  expressed  in  terms  of  u  with 
sufficient  accuracy  to  give  a  very  close  approximation  to  the 
value  of  p. 

For  values  of  u  not  too  remote  from  u%  the  function  r2  can  be 
expanded  in  a  converging  series  of  the  form 

(114)    r2  =  r22  +  ci(u  -  u2)  +  c^(u  -  uz)2  +  c3(u  -  w2)3  +  •  •  •  . 

In  an  unknown  orbit  the  coefficients  of  the  series  (114)  are 
unknown,  but  it  will  now  be  shown  how  a  sufficient  number  to 
define  p  with  the  desired  degree  of  accuracy  can  be  easily  found. 
By  hypothesis,  the  radii  and  arguments  of  latitude  are  known  at 
the  epochs  t\,  t2)  and  ts.  Hence  (114)  becomes  at  ti  and  t3 

*  F.  R.  Moulton;  The  Astronomical  Journal,  vol.  xxn.,  No.  510  (1901). 


139]  SECOND  METHOD   OF  DETERMINING   a,   6,   AND   03.  245 


r2 


(115) 


For  abbreviation  let 


(116) 


0-3 

€l 


Then  equations  (115)  can  be  written 


+  c2o-32  =  r  i2  —  r22  —  €1, 
=  r32  —  r22  —  e3. 


On  solving  for  c\  and  c3,  it  is  found  that 

-    61)(712  +    (f32   ~    63)(732   ~ 


Ci   = 


-  €3)0-3  - 


and,  on  substituting  the  values  of  ei  and  c3, 


(117) 


(7i(72(73 


7V0-2 


—  <73)  — 


Having  obtained  these  expressions  for  the  coefficients  of  the 
second  and  third  terms  of  (114),  let  this  series  be  substituted  for 
r2  in  (113)  and  the  result  integrated.  On  making  use  of  (116),  it  is 
easily  found  that 


<732) 


(733) 


246 


SECOND   METHOD   OF   DETERMINING   «,    6,    AND   CO. 


[139 


On  substituting  the  values  of  c\  and  c2  given  in  (117),  this  equation 
becomes 


(118) 


,         2/0 

r-  -E  --  (2o"3  - 


» 

00-10-3         00-3 


60-1 


x 

-  0-3) 


12 


30 


{4(0-3  —  o-i)2  +  0-10-3} 


If  the  second  observation  divides  the  whole  interval  into  two 
nearly  equal  parts,  as  generally  will  be  the  case  in  practice,  o-i 
and  0-3  will  be  nearly  equal.  Let 

(TI  —  0-3  =  e,     and     o-i  +  0-3  =  0"2j 


whence 


0-2 


0-3  = 


2 

0"2  —    € 


where  e  is  in  general  a  very  small   quantity.     On  substituting 
these  expressions  in  the  last  terms  of  (118)  this  equation  becomes 


(119) 


k 


/0 
(2o"3  - 


60-10-3  60-3 

+  -£&  (2o-i  -  0-3)  - 


It  is  found  in  a  similar  way  on  integrating  between  the  limits 
corresponding  to  tz  and  t\  that 

-2<rs) 


(120)    ^ 


, 


.x 
" 


For  the  intervals  of  time  which  are  used  in  determining  an 
orbit  these  series  converge  very  rapidly,  and  an  approximate  value 
of  p,  which  is  generally  as  accurate  as  is  desired,  can  be  obtained 


139]  SECOND   METHOD   OF   DETERMINING   a,   6  AND   CO.  247 

by  taking  only  the  first  three  terms*  in  the  right  member  of  (119). 
By  considering  equations  (119)  and  (120)  simultaneously  and 
neglecting  terms  in  c4  and  of  higher  order,  it  is  possible  to  deter- 
mine both  p  and  c3.  But  not  much  increase  in  accuracy  is  ob- 
tained because  the  term  in  c3  in  (119)  is  multiplied  by  the  small 
quantity  e,  while  that  in  c4  does  not  carry  this  factor.  Suppose 
the  value  of  p  has  been  computed;  it  will  be  shown  how  co  and  e 
can  be  found. 
The  polar  equation  of  the  conic  gives 

ff                  >.          P  —  7*1 
e  cos  (HI  —  co)  = , 
n 
t                     \           P   ~  r3 
e  cos  (us  —  co)  = . 
7*3 

Now  u$  —  co  —  (uz  —  HI)  +  (ui  —  co).  On  substituting  this  ex- 
pression for  us  —  co  in  the  second  equation  of  (121),  expanding, 
and  reducing  by  the  first,  it  is  found  that 


6  sin  (Ul  -  co)  =          ~  n   COS  U*  -  U     " 


(122) 

e  cos  (HI  —  co)  = -1 . 

Since  e  is  positive  these  equations  define  e  and  co  uniquely.  When 
p  and  e  are  known,  a  is  defined  by  p  =  a(l  —  e2)  or  p  =  a(e2  —  1) 
according  as  the  orbit  is  an  ellipse  or  an  hyperbola. 

If  the  elements  a,  e,  and  co  have  not  been  found  with  sufficient 
approximation  it  is  now  possible  to  correct  them.  It  follows  from 
(114)  that 


1  d3(r2)  1 

6  du23  '  ~  24 


and  since 


[1  +  e  cos  (u  —  co)]2 ' 
it  is  found  that 

*  For  conditions  and  rapidity  of  convergence  see  the  original  paper  in  the 
Astronomical  Journal,  No.  510.  It  is  shown  there  that  the  elements  of  asteroid 
orbits  will  be  given  by  the  first  three  terms  of  (119)  correct  to  the  sixth  decimal 
place  if  the  whole  interval  covered  by  the  observations  is  not  more -than 
40  days,  and  in  the  case  of  comets'  orbits,  if  the  interval  is  not  more  than  10 
days.  When  the  two  corrective  terms  defined  by  (123)  are  added  the  corre- 
sponding intervals  are  100  days  and  20  days. 


248        COMPUTATION   OF   THE    TIME.  OF   PERIHELION    PASSAGE.       [140 


(123) * 


e  sin  (u  —  co)  3e2  sin  (u  —  co)  cos  (u  —  co) 


3[1  +  e  cos  (u  —  co)]; 


—  e  cos 


—  co) 


[1  +  e  cos  (u  —  co)]4 

4e3  sin3  (^  —  co) 
[1  +  ecos  (w  -  co)]5' 

e2  sin2  (w  —  co) 
12[1  +  6  cos  (w  -  co)]3  ~~  [1  +  e  cos  (w  -  co)]4 

3e2  cos2  (M  -co)          6e3sin2(^-  co)  cos  (^  -  co) 
*"  4[1  +  e  cos  (u  -  co)]4  *       [1  +  e  cos  (u  -  co)]5 

,    5e4  sin4  (u  —  co) 

[1  +  e  cos  (w  —  co)]6' 

With  the  values  of  Cs  and  c±  computed  from  these  equations  the 
higher  terms  of  (119)  can  be  added,  thus  determining  a  more 
accurate  value  of  p,  after  which  e  and  co  can  be  recomputed  by 
(122).  Besides  being  very  brief  this  method  has  the  advantage  of 
being  the  same  for  all  conies. 

140.  Computation  of  the  Time  of  Perihelion  Passage.    The 

methods  of  computing  the  time  of  perihelion  passage  depend  upon 
whether  the  body  moves  in  a  parabola,  ellipse,  or  hyperbola,  and 
are  based  on  the  formulas  of  chap.  v. 

Parabolic  Case.     Equation  (32),  of  chap,  v.,  is 


(124) 


k(t  -  D  - 


where  2q  =  p.     Since  u  —  v  +  co,  and  HI,  w2,  and  us  are  known, 
this  equation  determines  T. 

Elliptic  Case.     The  first  two  equations  of  (49),  chap,  v.,  give 


(125) 


which  uniquely  define  E.     Then  Kepler's  equation 
(126)  M  =  n(t  -  T)  =  E  -  e  sin  E 

determines  T  by  using  v  and  the  corresponding  E  at  h,  t2,  or  t3. 
Hyperbolic  Case.     The  quantity  F  is  defined  by 


sin  E  = 

Vl  —  e2  sin  v 

\-\-e  cos  v 
e  +  cos  v 

\-\-e  cos  v  ' 

(127) 


-  1 


141]  DIRECT  DERIVATION    OF    EQUATIONS. 

after  which  T  is  given  by 
k  Vl  -f-  m 


249 


(128) 


(t  -  T)  =  -  F  +  e  sinh  F. 


141.  Direct  Derivation  of  Equations  Defining  Orbits.  The 
motion  of  an  observed  body  must  satisfy  both  geometrical  and 
dynamical  conditions.  Altogether  the  simplest  mode  of  pro- 
cedure is  to  write  out  at  once  these  conditions.  They  will  involve 
directly  or  indirectly  many  of  the  equations  of  the  methods  of 
Laplace  and  Gauss,  for  these  methods  both  rest  in  the  end  on  the 
essentials  of  the  problem. 

Let  the  notation  of  Art.  Ill  be  adopted.  Think  of  the  sun  as 
an  origin.  Then  obviously  the  ^-coordinate  of  C  equals  the 
^-coordinate  of  the  observer  plus  the  ^-coordinate  of  C  with  respect 
to  the  observer.  Similar  equations  are  of  course  true  in  the  two 
other  coordinates.  These  relations  are  explicitly 


-  \iPi 


a  =  i,  2, 3), 


(129) 


-f-  y*  =  - 


These  equations  are  subject  to  no  errors  of  parallax  because  the 
coordinates  of  the  observer  have  been  used.  Moreover,  they 
contain  all  the  geometrical  relations  which  exist  among  the  bodies 
S,  E,  and  C  at  h,  tz,  and  £3. 

The  next  condition  to  be  applied  is  that  C  shall  move  about  S 
according  to  the  law  of  gravitation.  This  is  equivalent  to  stating 
that  its  coordinates  can  be  developed  in  series  of  the  form  of  (73). 
On  making  use  of  this  notation,  equations  (129)  become 


(130) 


+  fiX0  + 


+  f&o  -f- 

+  /i2/o  + 

/z2p2  +  /22/o  +  £22/0' 


=  —  X2, 


=  —  Y 


+  fiZ0  +  g&o'  =  - 


—    V3P3   +  /320    +   QsZo'    =     —    Z3. 


250        FOBMULAS   FOE    COMPUTING  AN  APPROXIMATE   ORBIT.       [142 


If  the  date  of  the  second  observation  is  taken  as  the  origin  of 
time,  as  is  convenient  in  practice,  /2  =  1  and  gz  =  0. 

Equations  (130)  contain  fully  the  geometrical  and  dynamical 
conditions  of  the  problem  and  are  valid  for  all  classes  of  conies. 
Since  they  are  only  the  necessary  conditions  no  artificial  diffi- 
culties or  exceptional  cases  have  been  introduced;  and  if  in  a 
special  case  they  should  fail  no  other  mode  of  approach  could 
succeed. 

The  right  members  of  equations  (130)  are  entirely  known;  the 
unknowns  in  the  left  members  are  pi,  p2,  p3,  XQ,  XQ',  y0,  yQ',  z0,  and 
z</.  That  is,  the  number  of  unknowns  exactly  equals  the  number 
of  equations.  The  quantities  pi,  p2,  and  p3  enter  linearly,  but 
XQ,  •  •  • ,  2</  occur  not  only  explicitly  but  also  in  the  higher  terms 
of  the/*  and  the  gi.  The  solution  of  (130)  for  pi,  p2,  and  p3  is 


(131) 


where 


(132) 


A2pi  =  +  Ai  — 


A2p2  = 


+ 


r1  +  wi 


A2  = 


»*1;  A2;  AS 

Mi,  M2,  Ms 

^1,  ^2,  J'S 

Xi,  X»,  Xs 


Ml, 


J>3 


C2  +  C,, 
Xi,    X2, 

Xl,       X2, 

Mi,      M2, 

Vl,        VZ, 


M3 


In  order  to  complete  the  discussion  the  coefficients  of  the  deter- 
minants in  the  right  members  of  these  equations  must  be  developed, 
as  they  were  in  (86) ;  and  since  A2  is  of  the  third  order,  terms  of 
the  right  member  of  the  third  order  must  be  retained  even  in  the 
first  approximation.  When  applied  to  the  second  of  (131),  this 
leads  to  an  equation  of  the  form  of  the  first  of  (44).  The  details 
of  this  and  the  completion  of  the  solution  of  equations  (130)  will 
be  called  out  in  the  questions  which  follow  Art.  142. 

142.  Formulas  for  Computing  an  Approximate  Orbit.    For  con- 
venience in  use  the  formulas  for  the  computation  of  an  approxi- 


142]      FORMULAS   FOR   COMPUTING  AN  APPROXIMATE   ORBIT.         251 


mate  orbit  are  collected  here  in  the  order  in  which  they  are  used. 
The  numbers  attached  are  those  occurring  in  the  text. 

Preparation  of  the  data.  The  observed  right  ascensions  and 
declinations,  <*o  and  60,  are  corrected  for  precession,  aberration, 
etc.,  by 

ra  =  ao  —  15/  —  g  sin  (G  +  «o)  tan  50  —  h  sin  (H  -f  a0)  sec  60, 

(4)     4 

[  6  =  60  —  i  cos  60  —  g  cos  (G  +  ao)  —  h  cos  (#  +  a0)  sin  60. 
The  direction  cosines  are  given  by 

=  cos  dj  cos  a/,         (j  =  1,  2,  3), 


(ta-tb)(ta   ~   te) 


sn  a 


The  Method  of  Laplace.  Take  to  =  fa  unless  the  intervals 
between  the  successive  observations  are  very  unequal,  when 
to  =  K^i  +  ^2  -\-t  3).  It  will  be  supposed  that  tQ  =  fa-  Suppose  X, 
y,  and  Z  are  tabulated  in  the  Ephemeris  for  ta,  tb)  tc  where  tb  is 
near  tQ.  Then  compute  X,  Y,  and  Z  at  to  from  formulas  of  the 
type* 


(31) 

(26) 
(67) 

(64,  65) 

(67) 

(68) 


_i_     (^0     tg)(to     tb)    y, 

*"  (tc    -   ta)(te    ~   tbr 

k(ti  -  fa)  =  T,;        (j  =  1,  2,  3;  T2  =  0). 
P  =  —  nrz(rz  —  TI). 
Xi,     X2,     Xs 
Mi,     M2,      Ms 


T3 


Xi, 


MI,  M2, 


X 


*% 


=  +2-? 


Xi, 


MI,  M2, 


M2,      M3, 


X2, 


M2,        M3, 


X 
Y 
Z 
X 
Y 
Z 


*  These  equations  are  very  simple  because  ta,  tb,  and  tc  differ  by  intervals  of 
one  day,  but  there  are  other  methods  of  interpolation  which  are  even  simpler. 


252         FORMULAS   FOR   COMPUTING   AN   APPROXIMATE    ORBIT.       [142 

(46)          R  cos  }  =  X\  +  YH  +  Zv,     (0  <  ^  <L  TT). 


TV  sin  m  =  it  sin  ^, 

(47) 
(48) 

N  cosm  =  Rcost  -  ^  , 

sin4  ^>  =  Tkf  sin  ((p  +  m). 

Mtt                    =  Psin*               =/?sin(^+rf 

sm 


(44) 
(8) 


=- 

p  *      r3 


PX  - 


z  =  pv  —  Z 


Compute  X',  //,  ^'  from  equations  of  the  type 

^, ^(r2  +  r3)Xi 

(32)  (T1  " 


rs)        (r2  —  T3)(r2  — 
(TI  +  r2)X3 


(r3  —  TI)  (TS  —  T2) 
Compute  X',  Y',  and  Z'  from  equations  of  the  type 

(32)  (ta  ~ 


fe) 


y 


(8) 


PXr  - 


i  i        i  /  /7/ 

2!     =  p  j/  -j-  pj;     —   //  . 


At  this  point  the  correction  for  the  time  aberration  may  be 
made  by  equations  (70),  and  the  approximate  values  of  x,  y,  z, 
x'j  y',  and  z'  may  be  improved  by  the  methods  of  Arts.  128  and 
129;  or,  the  elements  may  be  computed  at  once  from  the  formulas 
given  in  chap.  v.  The  formulas  for  the  determination  of  the 
elements  will  be  given  and  the  numbers  of  the  equations  refer  to 
chap,  v 

The  integrals  of  areas  in  the  equator  system  are 


142]       FORMULAS   FOR    COMPUTING   AN   APPROXIMATE    ORBIT.         253 

)xy'  -  yx'  =  &i, 
yz'  -  zyf  =  &2, 
zx'  -  xzf  =  63. 

If  e  represents  the  obliquity  of  the  ecliptic,  the  corresponding 
constants  in  the  ecliptic  system  are 

1  =  61  cos  e  —  63  sin  e, 

2  =    &2, 

3  =  bi  sin  e  +  63  cos  e, 
and  i  and  ft  are  defined  by  (chap,  v.) 


(15) 


=   Vai2  +  a22  +  a32  cos  i, 

=  =*=  Vai2  +  a<>2  +  as2  sin  i  sin 


a3  =  q=   Vai2  +  a22  +  a32  sin  i  cos  ft. 
The  major  axis  and  parameter  are  defined  by 

(24)  x'2  +  y'2  +  z'2  =  1 

(22)  k*p  =  k2a(l  -  e2)  =  ax2  +  a22  +  a32. 

It  follows  from  Fig.  37,  p.  237,  that 

sin  i  sin  u  =  sin  b  =  - , 


11                 x 
cosi  sin  u  =  cos  6  sin  (I  —  ft)  =  -cos  ft sin  ft, 

77  /v» 

cos  u  =  cos  b  cos  (I  —  ft)  =  -sin  ft  H — cos  ft, 


which  define  u.    The  angle  v  is  given  by 


and 


I  +  e  cos  v ' 

0)   =   U  —  V. 

If  the  orbit  is  a  parabola,  T  is  defined  by 
(32)  k(t  -  T)  =  ip 


254         FORMULAS   FOR    COMPUTING   AN   APPROXIMATE   ORBIT.       [142 

If  the  orbit  is  an  ellipse,  E,  n,  and  T  are  denned  by 

\\  -  e,       v 


(50) 


E 
tan     = 


(30)  »-j, 

(42)  n(t  -  T)  =  E  -  e  sin  E. 

The  corresponding  equations  for  hyperbolic  orbits  are 

(73)  .  a  +  r  —  ae  cosh  F, 

(74)  n(t  -  T)  =  -  F  +  e  sinh  F. 

The  Method  of  Gauss.  The  observed  data  are  corrected  by  (4) 
and  the  direction  cosines  are  given  by  (6).  The  coordinates  of  the 
sun  at  tif  hi  and  h  can  be  computed  from  equations  of  the  type 


~   tb)(tj   -   tc)  (tj   - 

*        *" 


i   ~   tc) 


(ta  ~   tb)(ta  -te) 


(tb    ~    ta)(tb   ~    te) 


~   <a)fo   ~   fe)   y 

C 


(31) 


where  Xa,  ,Xb,  -X"c  are  taken  from  the  Ephemeris  and  tb  is  the  time 
nearest  to  ti  for  which  X  is  given.     Then 


(64) 


X2, 


(88) 


K   =- 


\i, 

Mi, 
^i, 

Xi, 

Mi, 


Xs  —  2X2, 
73  -  2F2, 


+  ^3,     Xs 
+  Fa,      Ms 


On  neglecting  the  last  term  of  (87),  which  is  very  small,  and 
comparing  the  result  with  the  first  of  (44),  it  is  seen  that  the 
explicit  formulas  for  determining  r2  and  p2  are 


(46) 


COS 


X,\t  +  ^2^2  +  Z,vt,          (0  <  fa  ^  IT), 


142]       FORMULAS  FOR   COMPUTING   AN   APPROXIMATE   ORBIT.         255 

C  N  sm  m  =  Rz  sin  ^2, 


(47) 

(48) 
(46) 


N  cos  m  =  Rz  cos  \f/z  —  T~  , 
M  =  — 2   J*m      >  0. 

sin4  <p  =  M  sin  (<p  +  m). 
7£  2  sin  1^2 


sin  <p 

=  p   sin  (fa  +  <?) 

sin  ^ 
Then  pi  and  p3  are  given  by 

_  [1,  3] 
Yi, 

_^  [1,  2] 
(89) 


Ml,     Ms 


PI  = 


+ 


[2,3] 


[2,3] 
-Xs,     X3 


Ms 


+  P2 


X3 
M3 

[1,3] 

[2,3] 


Xi,     X3 

Mi,     Ms 


P3    = 


[2,3] 


[1,2] 


X, 


Mi, 


[1,3] 
[1,2] 


+ 


Xi, 
Ml, 


+  P2 


Xi, 
Mi, 


[1,  3] 


[2,  3] 


X2,     X3 

M2,      Ms 


Xi,     X2 

Mi,      M2 


(or  by  formulas  obtained  from  these  by  cyclical  permutation  of 
the  letters  X,  /*,  v  and  X,  Y,  Z),  where 

(85)  2r  =  r3  - 

and 


(86) 


[1,3] 

[2, 3]       1 


2«  =  TS  +  TI, 
1 


[1,2] 
[2,3] 


[2,3] 
[1,2] 


[1,3] 
[1,2] 


2  T  2T  ^  4r23 

'7  +  27? 
1  +7  +  2r? 
1  +  -+^-: 


2r23 


±  +  ^- 
2      2r  "*"  4r23 


256         FORMULAS  FOR   COMPUTING  AN  APPROXIMATE   ORBIT.       [142 

=  P/Xy  -  Xit        (j  =  1,  2,  3), 
(8) 


At  this  point  the  correction  for  the  time  aberration  may  be 
made;  the  first  two  derivatives  of  r22  may  be  computed  from  the 
values  ri2,  r22,  and  r32  by  applying  the  formulas  (32)  to  this  case; 
p  and  q  may  be  computed  from  (74)  and  more  approximate 
values  of  P  and  Q  may  be  determined  from  (86);  and  then  the 
computation  may  be  repeated  beginning  with  equations  (46); 
or,  the  method  of  Gauss  of  Art.  134  may  be  used  to  improve  the 
accuracy  of  the  expressions  for  the  ratios  of  the  triangles;  or,  the 
elements  may  be  computed  without  further  approximation  of  the 
intermediate  quantities.  The  formulas  for  the  computation  of 
the  elements  will  be  given.  Let  the  rectangular  coordinates  in 
the  ecliptic  system  be  Xi,  y^  Zi,  and  the  obliquity  of  the  ecliptic  e, 
which  will  not  be  confused  with  the  e  defined  in  (85).  Then 

Xj,         (j  =  1,  2,  3), 

+  yj  cos  e  +  zi  sm  e> 
j  =  —  y}-  sin  e  +  Zj  cos  e. 

'  Ax,  +  By,  +  Czi  =  0, 
(17)  -   Ax,  +  By,  +  Cz2  =  0, 

-  Ax3  +  By,  +  Cz3  =  0, 


from  which 

A  :B  :C  = 


Vi, 


Then,  from  equations  corresponding  to  (11),  (14),  and  (15)  of 
chap,  v., 

A 


(15) 


sin 


cost  = 


sn  i 


cos  ft  sin  i  = 


VA2  +  B2  +  C2 ' 
=F  C 


which  define  ft  and  i. 

It  follows  from  Fig.  37  that  the  arguments  of  the  latitude  are 
defined  by 


PROBLEMS.  257 


sin  i  sin  Uj  =  -  ,         (j  =  1,  2,  3), 


v  •  X' 

cos  i  sin  Uj  =  —  cos  ft  —  —  sin 
TV  TV 


.  . 

=  —  sin  ft  H  —  -  cos  ft  . 
TV 


define  e  and  co.     Hence  a  can  be  determined  from  p  and  e. 

Since  w/  =  u,-  —  co  (j  =  1,  2,  3),  the  time  of  perihelion  passage 
is  determined  precisely  as  in  the  method  of  Laplace  by  equations 
(of  chap,  v.)  (32),  [(50),  (30),  (42)],  [(73),  (74)]  in  the  parabolic, 
elliptic,  and  hyperbolic  cases  respectively. 


XVII.     PROBLEMS. 

1.  Take  three  observations  of  an  asteroid  not  separated  from  one  another 
by  more  than  15  days,  or  three  of  a  comet  not  separated  from  one  another  by 
more  than  6  days,  and  compute  the  elements  of  the  orbit  by  both  the  method 
of  Laplace  and  also  that  of  Gauss. 

2.  Prove  that  the  apparent  motion  of  C  cannot  be  permanently  along  a 
great  circle  unless  it  moves  in  the  plane  of  the  ecliptic. 

3.  Apply  formulas  (31)  and  (32)  on  a  definite  closed  function,  as  for  ex- 
ample x  =  sin  t. 

4.  By  means  of  the  equation 

7-2  =  #2  _j_  p2  _  2Rr  cos  $ 

eliminate  p  from  the  first  equation  of  (44)  and  discuss  the  result  by  the  methods 
of  the  Theory  of  Algebraic  Equations,  and  show  that  the  solutions  agree 
qualitatively  with  those  obtained  in  Art.  119. 

5.  Discuss  the  determinants  D,  D\,  and  D2  when  there  are  four  observations. 

6.  Express  A2  when  there  are  three  observations  in  terms  of  the  on  and  the  5» 
in  such  a  manner  that  the  fact  it  is  of  the  third  order  will  be  explicitly  exhibited. 

18 


258  HISTORICAL   SKETCH. 

7.  Develop  the  explicit  formulas,  using  the  X;,  /*»-,  and  vi  and  the  determi- 
nant notation,  for  the  differential  corrections  of  the  method  of  Harzer  and 
Leuschner. 

8.  Give  a  geometrical  interpretation  of  the  vanishing  of  the  coefficients 
of  pi  and  ps  in  equations  (89). 

9.  Suppose  three  positions  of  C  are  known  as  in  Art.  139.     Show  (a)  that 
the  three  equations 

P  (1  =  1,2,3), 


1  +  e  cos  (in  —  co)  > 

define  p,  e,  and  co  without  using  the  intervals  of  time  in  which  the  arcs  are 
described;  (6)  write  out  the  explicit  formulas  for  computing  p,  e,  and  co; 
(c)  compare  then-  length  with  that  of  (119)  and  (122);  and  (d)  show  that  p  is 
not  well  determined  as  it  depends  upon  ratios  of  small  quantities  of  the  third 
order. 

10.  Suppose /2  =  1,  02  =  0  and  regard  (130)  as  linear  equations  in  pi,  p2,  p3, 
Xo,  XQ',  2/o,  2/o',  z0,  z0'.     Show  that  the  determinant  of  the  coefficients  is 


A    =    - 


Al, 
Ml, 
V\, 


11.  Show  that  on  using  the  expansions  of  equations  (86)  the  second  equa- 
tion of  (131)  becomes  (87). 

%  12.  Having  found  p2  from  the  equation  corresponding  to  (87),  and  p\  and  p3 
from  (131),  show  that  XO,'XQ',  yt,  y0',  20,  z</  can  be  determined  from  equations 
(130).  (Then  the  elements  can  be  determined  as  in  the  Laplacian  Method.) 


HISTORICAL  SKETCH  AND  BIBLIOGRAPHY. 

The  first  method  of  finding  the  orbit  of  a  body  (comet  moving  in  a  parab- 
ola) from  three  observations  was  devised  by  Newton,  and  is  given  in  the 
Principia,  Book  in.,  Prop.  XLI.  The  solution  depends  upon  a  graphical  con- 
struction, which,  by  successive  approximations,  leads  to  the  elements.  One 
of  the  earliest  applications  of  the  method  was  by  Halley  to  the  comet  which 
has  since  borne  his  name.  Newton  seems  to  have  had  trouble  with  the 
problem  of  determining  orbits,  for  he  said,  "  This  being  a  problem  of  very 
great  difficulty,  I  tried  many  methods  of  resolving  it."  Newton's  success  in 
basing  his  discussion  on  the  fundamental  elements  of  the  problem  was  fully 
explained  by  Laplace  in  his  memoir  on  the  subject 

The  first  complete  solution  which  did  not  depend  upon  a  graphical  con- 
struction was  given  by  Euler  in  1744  in  his  Theoria  Motuum  Planetarum  et 
Cometarum.  Some  important  advances  were  made  by  Lambert  in  1761. 
Up  to  this  time  the  methods  were  for  the  most  part  based  upon  one  or  the 
other  of  two  assumptions,  which  are  only  approximately  true,  viz.,  that  in 
the  interval  t3  —  ti  the  observed  body  describes  a  straight  line  with  uniform 
speed,  or  that  the  radius  at  the  time  of  the  second  observation  divides  the 


HISTORICAL   SKETCH.  259 

chord  joining  the  end  positions  into  segments  which  are  proportional  to  the 
intervals  between  the  observations.  In  attempting  to  improve  on  the  second 
of  these  assumptions  Lambert  made  the  discovery  of  the  relation  among  the 
radii,  chord,  time-interval,  and  major  axis  mentioned  in  Art.  92.  He  later 
made  the  determination  depend  upon  the  curvature  of  the  apparent  orbit, 
which  is  closely  related  to  the  determinant  A2,  and  in  this  direction  approached 
the  best  modern  methods.  He  had  an  unusual  grasp  of  the  physics  and 
geometry  of  the  problem,  and  really  anticipated  many  of  the  ideas  which 
were  carried  out  by  his  successors  in  better  and  more  convenient  ways. 

Lagrange  wrote  three  memoirs  on  the  theory  of  orbits,  two  in  1778  and 
one  in  1783.  They  are  printed  together  in  his  collected  works,  vol.  iv.,  pp.  439- 
532.  As  one  would  expect,  with  Lagrange  came  generality,  precision,  and 
mathematical  elegance.  He  determined  the  geocentric  distance  of  C  at  the 
time  of  the  second  observation  by  an  equation  of  the  eighth  degree,  which 
is  nothing  else  than  (87)  with  r-i  eliminated  by  means  of  the  equation  which 
expresses  the  fact  that  S,  E,  and  C  form  a  triangle  at  t-i.  He  developed  the 
expressions  for  the  heliocentric  coordinates  as  power  series  in  the  time-intervals 
[eqs.  (73)],  and  laid  the  foundation  for  the  development  of  expressions  for 
intermediate  elements  in  power  series.  These  developments  have  been  com- 
pleted and  put  in  form  for  numerical  applications  by  Charlier,  Meddelande 
frdn  Lunds  A  stronomiska  Observatorium,  No.  46.  The  original  work  of 
Lagrange  was  not  put  in  a  form  adapted  to  the  needs  of  the  computer,  and 
has  not  been  used  in  practice. 

In  1780  Laplace  published  an  entirely  new  method  in  Memoires  de  V Acad- 
emic Royale  des  Sciences  de  Paris  (Collected  Works,  vol.  x.,  pp.  93-146).  This 
method,  the  fundamental  ideas  of  which  have  been  given  in  this  chapter,  has 
been  the  basis  for  a  great  deal  of  later  work.  Among  the  developments  in 
this  line  may  be  mentioned  a  memoir  by  Villarceau  (Annales  de  I'Observa- 
toire  de  Paris,  vol.  in.),  the  work  of  Harzer  (Astronomische  Nachrichten,  vol. 
141),  and  its  simplification  by  Leuschner  (Publications  of  the  Lick  Observa- 
tory, vol.  vii.,  Part  i.).  The  approximations  beyond  the  first  are  not  con- 
veniently carried  out  in  the  original  method  of  Laplace,  but  the  method  of 
differential  corrections  devised  by  Harzer  and  simplified  by  Leuschner  has 
proved  very  satisfactory  in  practice. 

Olbers  published  his  classical  Abhandlung  uber  die  leichteste  und  bequemste 
Methode,  die  Bahn  eines  Kometen  zu  berechnen,  in  1797.  This  method  has  not 
been  surpassed  for  computing  parabolic  orbits  and  is  in  very  general  use  even 
at  the  present  time.  It  is  given  in  nearly  every  treatise  on  the  theory  of 
determining  orbits. 

The  discovery  of  Ceres  in  1801  and  its  loss  after  having  been  observed  only 
a  short  time  drew  the  attention  of  a  brilliant  young  German  mathematician, 
Gauss,  to  the  problem  of  determining  the  elements  of  the  orbit  of  a  heavenly 
body  from  observations  made  from  the  earth.  The  problem  was  quickly 
solved,  and  an  application  of  the  method  led  to  the  recovery  of  Ceres.  Gauss 
elaborated  and  perfected  his  work,  and  in  1809  brought  it  out  in  his  Theoria 
Motus  Corporum  Coeleslium.  This  work,  written  by  a  man  at  once  a  master 
of  mathematics  and  highly  skilled  as  a  computer,  is  so  filled  with  valuable 
ideas  and  is  so  exhaustive  that  it  remains  a  classic  treatise  on  the  subject  to 
this  day.  The  later  treatises  all  are  under  the  greatest  obligations  to  the  work 
of  Gauss. 


260  HISTORICAL   SKETCH. 

In  the  Memoirs  of  the  National  Academy  of  Science,  vol.  iv.  (1888),  Gibbs 
published  a  method  of  considerable  originality  in  which  the  first  approximation 
to  the  ratios  of  the  triangles  was  obtained  more  exactly  by  including  all  three 
geocentric  distances  as  unknown  from  the  beginning.  The  method  is  also 
distinguished  by  the  fact  that  it  was  developed  by  the  calculus  of  vector 
analysis. 

The  works  to  be  consulted  are: 

The  Theoria  Motus  of  Gauss. 

Watson's  Theoretical  Astronomy  (now  out  of  print). 

Oppolzer's  Bahnbestimmung,  an  exhaustive  treatise. 

Tisserand's  Legons  sur  la  Determination  des  Orbites,  written  in  the  char- 
acteristically clear  French  style. 

Bauschinger's  Bahnbestimmung,  a  recent  book  of  great  excellence  by  one 
of  the  best  authorities  on  the  subject  of  the  theory  of  orbits. 

Klinkerfues'  Theoretische  Astronomie  (third  edition  by  Buchholz),  an 
excellent  work  and  the  most  exhaustive  one  yet  issued. 


CHAPTER  VII. 

THE  GENERAL  INTEGRALS  OF  THE  PROBLEM  OF  n  BODIES. 

143.  The  Differential  Equations  of  Motion.  Suppose  the 
bodies  are  homogeneous  in  spherical  layers;  then  they  will  attract 
each  other  as  though  their  masses  were  at  their  centers.  Let  mi, 
^2,  •  •  •  ,  mn  represent  their  masses.  Let  the  coordinates  of  m< 
'referred  to  a  fixed  system  of  axes  be  Xi,  yi,  Z{  (i  =  1,  •  •  •,  n).  Let 
rt-,  /  represent  the  distance  between  the  centers  of  mt-  and  m/. 
Let  k2  represent  a  constant  depending  upon  the  units  employed. 
Then  the  components  of  force  on  mi  parallel  to  the  z-axis  are 

(xi  —  xn) 


r\2  ri,2  r\n 

and  the  total  force  is  their  sum.     Therefore 

(xi  —  x,) 


and  there  are  corresponding  equations  in  y  and  z. 

There  are  similar  equations  for  each  body,  and  the  whole  system 
of  equations  is 

d2Xi  n        f~       "x 


(i) 


m 

m,- 


Each  of  these  equations  involves  all  of  the  3n  variables  re,-,  7/i, 
and  Zi,  and  the  system  must,  therefore,  be  solved  simultaneously. 
There  are  3n  equations  each  of  the  second  order,  so  that  the 
problem  is  of  order  6n. 

Equations  (1)  can  be  put  in  a  simple  and  elegant  form  by  the 
introduction  of  the  potential  function,  which  in  this  problem  will 
be  denoted  by  U  instead  of  V.  The  constant  k2  will  be  included 
in  the  potential.  In  chap,  iv  the  potential,  7,  was  defined  by 

261  ' 


262  SIX  INTEGRALS   OF  MOTION   OF   CENTER   OF  MASS.  [144 

the  integral  V  =   I  —  .     In  this  case  the  system  is  composed  of 
J    p 


p 
discrete  masses,  and  the  potential  is 

(2)          v  -&?* 


The  partial  derivative  of  U  with  respect  to  xt  is 

dU      79       d    A  mj  79      A       (xt  —  xj) 

~~       ~  -  - 


and  there  are  similar  equations  in  ?/t-  and  2,- .     Therefore  equations 
(1)  can  be  written  in  the  form 


dU 

(3)  """      dU 

dU 


(i  =  1,   •••,  n). 

144.  The  Six  Integrals  of  the  Motion  of  the  Center  of  Mass. 

The  function  U  is  independent  of  the  choice  of  the  coordinate 
axes  since  it  depends  upon  the  mutual  distances  of  the  bodies 
alone.  Therefore,  if  the  origin  is  displaced  parallel  to  the  #-axis 
in  the  negative  direction  through  a  distance  a,  the  x-coordinate 
of  every  body  will  be  increased  by  the  quantity  a,  but  the  potential 
function  will  not  be  changed.  Let  the  fact  that  U  is  a  function 
of  all  the  x-coordinates  be  indicated  by  writing 

U  =    U(X19    X2,     -..,    Xn). 

After  the  origin  is  displaced  the  ^-coordinates  become 

Xi'  =  Xi  +  a,     (i  =  1,   •••,  n). 
The  partial  derivative  of  U  with  respect  to  a  is 

dU  =  dU^  dxi'       d£7  dxj  dU  dxnf 

da        dx\    da        dxz'  da  dxnr   da 

But  -r-1-  =  1,  (i  =  1,  •  •  •    n),  and  -r-  =  0,  because  U  does  not 
ua  da 

involve  a  explicitly.     Therefore,   on  dropping  the  accents  and 


144]  SIX   INTEGRALS    OF   MOTION    OF    CENTER    OF   MASS.  263 


writing  the  corresponding  equations  in  ?/»  and  z»  for  displacements 
j8  and  7,  it  is  found  that 


da 

dlL 
d(3 


Therefore  equations  (3)  give 


— *  =  0 

dp    u- 


These  equations  are  at  once  integrable,  and  the  result  of  inte-x 
gration  is 


(4) 


where  «i,  j8i,  71  are  the  constants  of  integration.     On  integrating 
again,  it  follows  that 


(5) 


Let  J^  m{  =  M,  and  5,  y,  and  2  represent  the  coordinates  of 
the  center  of  mass  of  the  n  bodies;  then,  by  Art.  19, 


264  THE   THREE   INTEGRALS   OF   AREAS.  [145 

i  Xi  =  MX, 

n 

(6) 


iZi  =  Mz. 

Therefore,  equations  (5)  become 

IMx  =  out  +  012, 
Aff-jM-hft, 
Mz  =  7i*  +  72; 

that  is,  the  coordinates  of  the  center  of  mass  vary  directly  as  the 
time.  From  this  it  can  be  inferred  that  the  center  of  mass  moves 
with  uniform  speed  in  a  straight  line.  Or  otherwise,  the  velocity 
of  the  center  of  mass  is 


which  is  a  constant;  and  on  eliminating  t  from  equations  (7),  it 
is  found  that 

MX  —  a*      My  —  02      Mz  —  72 


(9) 


01  7i 


which  are  the  symmetrical  equations  of  a  straight  line  in  space  of 
three  dimensions.  Equations  (8)  and  (9)  give  the  theorem: 

//  n  bodies  are  subject  to  no  forces  except  their  mutual  attractions, 
their  center  of  mass  moves  in  a  straight  line  with  uniform  speed. 
The  special  case  V  =  0  will  arise  if  «i  =  0i  =  71  =  0.  Since  it 
is  impossible  to  know  any  fixed  point  in  space  it  is  impossible 
to  determine  the  six  constants. 

The  origin  might  now  be  transferred  to  the  center  of  mass  of 
the  system,  as  it  was  in  the  Problem  of  Two  Bodies,  or,  to  the 
center  of  one  of  the  bodies,  as  it  will  be  in  Art.  148,  and  the  order 
of  the  problem  reduced  six  units. 

145.  The  Three  Integrals  of  Areas.  The  potential  function  is 
noli,  changed  by  a  rotation  of  the  axes.  Suppose  the  system  of 
coordinates  is  rotated  around  the  z-axis  through  the  angle  —  </>, 
and  call  the  new  coordinates  x/,  ?//,  and  z/.  They  are  related  to 
the  old  by  the  equations 


145] 
(10) 


THE  THREE  INTEGRALS  OF  AREAS. 


i  =  Xi  cos  0  —  7/t  sin 


265 


/  =  Xi  sin  <£  +  i/i  cos  $, 
/  =  Zi,         (i  =  1,  •••,.*)• 

Since  the  function  U  is  not  changed  by  the  rotation  it  does  not 
contain  0  explicitly;  therefore 

ATT   Av  '  w     AT7    %»,  '  n     ATT    A»  S 

OU    OXi 


(ID     ¥  = 


But  from  (10)  it  follows  that 
therefore  (11)  becomes 


= 
* 


(t  =  1,   •••,  ri); 


On  dropping  the  accents,  which  are  of  no  further  use,  it  is  found 
as  a  consequence  of  (3)  that 


F      d2?/i          d2X{"|       _ 
111*  I       *  r  -^"3^      = 


,    .    .,    , 
similarly, 


Each  term  of  these  sums  can  be  integrated  separately,  giving 


(12) 


The  parentheses  are  the  projections  of  the  areal  velocities  of  the 
various  bodies  upon  the  three  fundamental  planes  (Art.  16). 
As  it  is  impossible  to  determine  any  fixed  point  in  space,  so  also 
it  is  impossible  to  determine  any  fixed  direction  in  space;  conse- 
quently it  is  impossible  to  determine  practically  the  constants 
Ci,  c2,  c3.  Yet,  in  this  case  it  is  customary  to  assume  that  the 


266 


THE    THREE   INTEGRALS    OF   AREAS. 


[145 


fixed  stars,  on  the  average,  do  not  revolve  in  space,  so  that,  by 
observing  them,  these  constants  can  be  determined.  It  is  evident, 
however,  that  there  is  no  more  reason  for  assuming  that  the  stars 
do  not  revolve  than  there  is  for  assuming  that  they  are  not  drifting 
through  space,  each  being  a  pure  assumption  without  any  possi- 
bility of  proof  or  disproof.  But  it  is  to  be  noted  that,  if  these 
assumptions  are  granted,  the  constants  ci,  c2,  and  c3  can  be  deter- 
mined easily  with  a  high  degree  of  precision,  while  in  the  present 
state  of  observational  Astronomy  the  constants  of  equations  (4) 
cannot  be  found  with  any  considerable  accuracy. 

Let  Ai,  Bi,  and  d  represent  the  projections  of  the  areas  de- 
scribed by  the  line  from  the  origin  to  the  body  mi  upon  the  xy,  yz, 
and  20>planes  respectively;  then  (12)  can  be  written 


dBi 


dd 
dt 


=   C3, 


the  integrals  of  which  are 


(13) 


6f 

n 


Ci', 


c3'. 


^ 


Hence  the  theorem: 

The  sums  of  the  products  of  the  masses  and  the  projections  of  the 
areas  described  by  the  corresponding  radii  are  proportional  to  the 
time;  or,  from  (12),  the  sums  of  the  products  of  the  masses  and 
the  rates  of  the  projections  of  the  areas  are  constants. 

It  is  possible,  as  was  first  shown  by  Laplace,  to  direct  the  axes 
so  that  two  of  the  constants  in  equations  (12)  shall  be  zero,  while 
the  third  becomes  Vci2  +  c22  -+-  c32.  This  is  the  plane  of  maxi- 
mum sum  of  the  products  of  the  masses  and  the  rates  of  the  pro- 
jections of  areas.  Its  relations  to  the  original  fixed  axes  are 
defined  by  the  constants  ci,  c2,  c3,  and  its  position  is,  therefore, 
always  the  same.  On  this  account  it  was  called  the  invariable 


146]  THE  ENERGY  INTEGRAL.  267 

plane  by  Laplace.  At  present  the  invariable  plane  of  the  solar 
system  is  inclined  to  the  ecliptic  by  about  2°,  and  the  longitude 
of  its  ascending  node  is  about  286°.  These  figures  are  subject  to 
some  uncertainty  because  of  our  imperfect  knowledge  regarding 
the  masses  of  some  of  the  planets.  If  the  position  of  the  plane 
were  known  with  exactness  it  would  possess  some  practical  ad- 
vantages over  the  ecliptic,  which  undergoes  considerable  vari- 
ations, as  a  fundamental  plane  of  reference.  It  has  been  of  great 
value  in  certain  theoretical  investigations.* 

146.  The  Energy  Integral.f  On  multiplying  equations  (3)  by 
-~ ,  -jf ,  -jj  respectively,  adding,  and  summing  with  respect  to  i, 
it  is  found  that 


li  {  dt2  dt  +  dt2  dt  "h  dt2  dt  J 


/i    .    acy  aY/t       ou  azi  i 
t4i  I  dxi  dt       dyt  dt       dzt  dt  )' 

The  potential  U  is  a  function  of  the  3n  variables  a;*-,  ?/;,  zi}  alone; 
therefore  the  right  member  of  (14)  is  the  total  derivative  of  U 
with  respect  to  t.  Upon  integrating  both  members  of  this  equa- 
tion, it  is  found  that 

(15)      \ 

The  left  member  of  this  equation  is  the  kinetic  energy  of  the  whole 
system,  and  the  right  member  is  the  potential  function  plus  a 
constant. 

Let  the  potential  energy  of  one  configuration  of  a  system  with 
respect  to  another  configuration  be  defined  as  the  amount  of  work 
required  to  change  it  from  the  one  to  the  other.  If  two  bodies 
attract  each  other  according  to  the  law  of  the  inverse  squares,  the 

force  existing  between  them  is  — r-5— ' .     The  amount  of  work  done 

ft.  i 

in  changing  their  distance  apart  from  r(9^  to  rt-,  /  is 
(16) 


*  See  memoirs  by  Jacobi,  Journal  de  Math.,  vol.  ix.;  Tisserand,  M6c.  Cbl. 
vol.  i.,  chap,  xxv.;  Poincare",  Les  Methodes  Nouvelles  de  la  Mec.  Cel.,  vol.  i., 
p.  39. 

t  This  is  very  frequently  called  the  Vis  Viva  integral. 


268  THE    QUESTION   OF  NEW  INTEGRALS.  [147 

If  the  bodies  are  at  an  infinite  distance  from  one  another  at  the 
start,  then  K0),  =  °o,  and  (16)  becomes 


hence 


Therefore,  17  is  the  negative  of  the  potential  energy  of  the  whole 
system  with  respect  to  the  infinite  separation  of  the  bodies  as  the 
original  configuration.  Hence  (15)  gives  the  theorem: 

In  a  system  of  n  bodies  subject  to  no  forces  except  their  mutual 
attractions  the  sum  of  the  kinetic  and  potential  energies  is  a  constant. 

147.  The  Question  of  New  Integrals.  Ten  of  the  whole  6n 
integrals  which  are  required  in  order  to  solve  the  problem  com- 
pletely have  been  found.  These  ten  integrals  are  the  only  ones 
known,  and  the  question  arises  whether  any  more  of  certain  types 
exist. 

In  a  profound  memoir  in  the  Ada  Mathematica,  vol.  xi.,  Bruns 
has  demonstrated  that,  when  the  rectangular  coordinates  are 
chosen  as  dependent  variables,  there  are  no  new  algebraic  integrals. 
This  does  not,  of  course,  exclude  the  possibility  of  algebraic  inte- 
grals when  other  variables  are  used.  Poincare'  has  demonstrated 
in  his  prize  memoir  in  the  Ada  Mathematica,  vol.  xin.,  and  again 
with  some  additions  in  Les  Methodes  Nouvelles  de  la  Mecanique 
Celeste,  chap,  v.,  that  the  Problem  of  Three  Bodies  admits  no  new 
uniform  transcendental  integrals,  even  when  the  masses  of  two 
of  the  bodies  are  very  small  compared  to  that  of  the  third.  In  this 
theorem  the  dependent  variables  are  the  elements  of  the  orbits 
of  the  bodies,  which  continually  change  under  their  mutual 
attractions.  It  does  not  follow  that  integrals  of  the  class  con- 
sidered by  Poincare*  do  not  exist  when  other  dependent  variables 
are  employed.  In  fact,  Levi-Civita  has  shown  the  existence  of 
this  class  of  integrals  in  a  special  problem,  which  comes  under 
Poincare*  's  theorem,  when  suitable  variables  are  used  (Ada 
Mathematica,  vol.  xxx.).  The  practical  importance  of  the 
theorems  of  Bruns  and  Poincare*  have  often  been  overrated  by 
those  who  have  forgotten  the  conditions  under  which  they  have 
been  proved  to  hold  true. 


148]  TRANSFER  OF   ORIGIN   TO   THE   SUN.  269 


XVIH.    PROBLEMS. 

1.  Write  equations  (1)  when  the  force  varies  inversely  as  the  nth  power 
of  the  distance.     For  what  values  of  n  do  the  equations  all  become  inde- 
pendent?    The  Problem  of  n  Bodies  can  be  completely  solved  for  this  law 
of  force;  show  that  the  orbits  with  respect  to  the  center  of  mass  of  the  system 
are  all  ellipses  with  this  point  as  center.     Show  that  the  orbit  of  any  body 
with  respect  to  any  other  is  also  a  central  ellipse,  and  that  the  same  is  true 
for  the  motion  of  any  body  with  respect  to  the  center  of  mass  of  any  sub- 
group of  the  whole  system.     Show  that  the  periods  are  all  equal. 

2.  What  will  be  the  definition  of  the  potential  function  when  the  force 
varies  inversely  as  the  nth  power  of  the  distance? 

3.  Derive  the  equations  immediately  preceding  (4)  directly  from  equa- 
tions (1). 

4.  Prove  that  the  theorem  regarding  the  motion  of  the  center  of  mass  holds 
when  the  force  varies  as  any  power  of  the  distance. 

5.  Derive  the  equations  immediately  preceding  (12)  directly  from  equa- 
tions (1),  and  show  that  they  hold  when  the  force  varies  as  any  power  of  the 
distance. 

6.  Any  plane  through  the  origin  can  be  changed  into  any  other  plane 
through  the  origin  by  a  rotation  around  each  of  two  of  the  coordinate  axes. 
Transform  equations  (12)  by  successive  rotations  around  two  of  the  axes,  and 
show  that  the  angles  of  rotation  can  be  so  chosen  that  two  of  the  constants, 
to  which  the  functions  of  the  new  coordinates  similar  to  (12)  are  equal,  are 
zero,  and  that  the  third  is  V  Ci2  +  c22  +  c32.     (This  is  the  method  used  by 
Laplace  to  prove  the  existence  of  the  invariable  plane.) 

7.  Why  are  equations  (13)  not  to  be  regarded  as  integrals  of  the  differ- 
ential equations  (1),  thus  making  the  whole  number  of  integrals  thirteen? 

148.  Transfer  of  the  Origin  to  the  Sun.  Nothing  is  known  of 
the  absolute  motions  of  the  planets  because  the  observations 
furnish  information  regarding  only  their  relative  positions,  or 
their  positions  with  respect  to  the  sun.  It  is  true  that  it  is  known 
that  the  solar  system  is  moving  toward  the  constellation  Hercules, 
but  it  must  be  remembered  that  this  motion  is  only  with  respect 
to  certain  of  the  stars.  The  problem  for  the  student  of  Celestial 
Mechanics  is  to  determine  the  relative  positions  of  the  members 
of  the  solar  system;  or,  in  particular,  to  determine  the  positions 
of  the  planets  with  respect  to  the  sun.  To  do  this  it  is  advanta- 
geous to  transfer  the  origin  to  the  sun,  and  to  employ  the  resulting 
differential  equations. 


270 


TRANSFER  OF   ORIGIN   TO   THE   SUN. 


[148 


Suppose  mn  is  the  sun  and  take  its  center  as  the  origin,  and  let 
the  coordinates  of  the  body  mi  referred  to  the  new  system  be 
Xi,  i//,  Zi.  Then  the  old  coordinates  are  expressed  in  terms  of 
the  new  by  the  equations 

Xi  =  Xi  +  xn,    yi  =  y*'  +  yn,    Zi  =  Zi  +  z»,     (i  =  l,  ••••,  n-1). 

Since  the  differences  of  the  old  variables  are  equal  to  the  corre- 
sponding differences  of  the  new,  it  follows  that 


3UdU 


dxi      dxS'         dyi      dyr         dZi      dzi" 

As  a  consequence  of  these  transformations  equations  (3)  become 


(17) 


d2Xjf      d2xn  _  J_  dU 
~W  '  ~  ~<W   ~  mi  3x7  ' 


^  + 


d2yrt 
dt2 


dt2 


I-  + 


dt2 


1  dU 

mi  dyt' ' 

1  dU 

mi  dz^ 


=  1,   •-.,  n-  1). 


Since  the  origin  is  at  xn'  =  yn'  =  zn'  =  0,  the  first  equation  of 
(1)  gives,  on  putting  i  =  n, 


/10\ 
(lo) 


d?xn  = 

rift     '         r3,        " 
'    1,  n 


. 


« 

2,  n 


n—  1,  n 


This  equation,  with  the  corresponding  ones  in  y  and  z,  substituted 
in  (17)  completes  the  transformation  to  the  new  variables;  but 
it  will  be  advantageous  to  combine  the  terms  in  another  manner 
so  that  those  which  come  from  the  attraction  of  the  sun  shall  be 
separate  from  the  others.  The  differential  equations  will  be 
written  for  the  body  m\,  from  which  the  others  can  be  formed  by 
permuting  the  subscripts. 

The  potential  function  can  be  broken  up  into  the  sum 


U 


or. 


si  rt-, 


n  —  1  n  —  1 


i  *  j); 


(19) 


U 


U'. 


149] 


DYNAMICAL   MEANING    OF    THE   EQUATIONS. 


271 


On  substituting  equations  (18)  and  (19)  in  equations  (17),  the 
latter  become 


(20)    < 


Let 


xi 


dt2 


m«) 


1   dU' 


mi 


n,,- 


then,  equations  (20)  can  be  written  in  the  form 


(21) 


'1, 


Let  the  accents,  which  have  become  useless,  be  dropped,  and, 
in  order  to  derive  the  general  equations  corresponding  to  (21),  let 


(22) 


Then,  the  general  equations  for  relative  motion  are 


(23) 


dt* 


. 
i  +  m,») 


xt 


=  >   my 


dt*  nji*i,n- 

in  which  i  =  1,  •  •  •,  n  —  1. 

149.  Dynamical  Meaning  of  the  Equations.  In  order  to  under- 
stand easily  the  meaning  of  the  equations,  suppose  that  there  are 
but  three  bodies,  mi,  m2,  and  mn  Suppose  mn  is  the  sun,  let  its 
mass  equal  unity,  and  let  the  distances  from  it  to  mi  and  m2  be 
r\  and  r2  respectively.  Then  equations  (23)  are,  in  full, 


272 


DYNAMICAL   MEANING   OF   THE    EQUATIONS. 


— 


x\ 
-3 


A.JJL 

dxi  \  ri,  2 


[149 

}. 


r2, 


dt2 

If  m2  were  zero  the  first  three  equations  would  be  independent 
of  the  second  three,  and  they  would  then  be  the  equations  for  the 
relative  motion  of  the  body  mi  with  respect  to  mn  =  1,  and  could 
be  integrated.  All  the  variations  from  the  purely  elliptical 
motion  arise  from  the  presence  of  the  right  members,  which  are, 
in  the  first  three  equations,  the  partial  derivatives  of  RI,  2  with 
respect  to  the  variables  xi,  yi,  and  z\  respectively.  On  this  account 
mzRi,  2  is  called  the  perturbative  function. 

The  partial  derivatives  of  the  first  terms  of  the  right  members 
of  the  first  three  equations  are  respectively 

(zi  -  z2) 


1,  2 


which  are  the  components  of  acceleration  of  nil  due  to  the  attrac 
tion  of  w2.     The  partial  derivatives  of  the  second  terms  are 


which  are  the  negatives  of  the  components  of  the  acceleration  of 
the  sun  due  to  the  attraction  of  w2.  Therefore  the  right  members 
of  the  first  three  equations  of  (24)  are  the  differences  of  the  com- 
ponents of  acceleration  of  mi  and  of  the  sun  due  to  the  attraction 
of  m2.  Similarly,  the  right  members  of  the  last  three  equations 
are  the  differences  of  the  components  of  the  acceleration  of  w2 
and  of  the  sun  due  to  the  attraction  of  mi.  If  two  bodies  are 
subject  to  equal  parallel  accelerations  their  relative  positions  will 
not  be  changed.  The  differences  of  their  accelerations  are  due  to 


150]  THE    ORDER   OF   THE    SYSTEM   OF   EQUATIONS.  273 

the  disturbing  forces,  and  measure  these  disturbances.  The  right 
members  of  (24)  are,  therefore,  exactly  those  parts  of  the  accelera- 
tions due  to  the  disturbing  forces. 

If  there  are  n  —  2  disturbing  bodies  the  right  members  are  the 
sums  of  terms  depending  upon  the  bodies  w2,  •  ••,  mn_i  similar  to 
the  right  members  of  (24),  which  depend  upon  ra2  alone;  or,  in 
other  words,  the  whole  resultants  of  the  disturbing  accelerations 
are  equal  to  the  sums  of  the  parts  arising  from  the  action  of  the 
separate  disturbing  bodies. 

150.  The  Order  of  the  System  of  Equations.  The  order  of  the 
system  of  equations  (23)  is  6n  —  6,  instead  of  6n  as  (1)  was  in 
the  case  of  absolute  motion.  In  the  absolute  motion  ten  integrals 
were  found  which  reduced  the  problem  to  order  Qn  —  10.  Six  of 
these  related  to  the  motion  of  the  center  of  mass,  three  to  the 
areal  velocities,  and  one  to  the  energy  of  the  system.  In  the 
present  case  but  four  integrals,  the  three  integrals  of  areas  and  the 
energy  integral,  can  be  found,  which  leaves  the  problem  of  order 
6n  —  10  also. 

The  problem  can  be  reduced  to  the  order  6n  —  6  by  using  the 
integrals  for  the  center  of  mass  directly.  In  particular,  consider 
the  differential  equations  for  the  bodies  mi,  w2,  •  •  •  ,  wn_i.  In  the 
original  equations  they  involve  the  coordinates  of  mn,  but  these 
quantities  can  be  eliminated  by  means  of  equations  (5). 

If  the  origin  is  taken  at  the  center  of  mass 

n  n  n 

t-2/»  =  0,  ra^Zi  =  0, 


and  the  elimination  becomes  particularly  simple.  Or,  because  of 
these  linear  homogeneous  relations,  the  n  variables  of  each  set 
can  be  expressed  linearly  and  homogeneously  in  terms  of  n  —  1 
new  variables.  Thus 


-l£n- 


CLn22  •  •  •  O,n,  B-ln-l, 

and  similar  sets  of  equations  for  y  and  z.  The  coefficients  a#  are 
arbitrary  constants  except  that  they  must  be  so  chosen  that  every 
determinant  of  the  matrix  of  the  substitutions  shall  be  distinct 
from  zero;  for,  otherwise,  a  linear  relation  would  exist  among  the  &. 
These  constants  can  be  so  chosen  that  the  transformed  equations 
19 


274  PROBLEMS. 

preserve  a  symmetrical  form.  This  method  was  employed  by 
Jacobi  in  an  important  memoir  entitled,  Sur  I'elimination  des 
noeuds  dans  le  probleme  des  trois  corps  (Journal  de  Math.  vol.  ix., 
1844),  and  by  Radau  in  a  memoir  entitled,  Sur  une  transformation 
des  equations  differentielles  de  la  Dynamique  (Annales  de  I' E  cole 
Normale,  1st  series,  vol.  v.). 


XIX.     PROBLEMS. 

1.  Make  the  transformation  xt  =  Xi   +  xn  in  the  integrals  (12)  and  (15), 
and  eliminate  xn,  yn,  zn,  -V^,  -J^,  and  -~  by  means  of  equations  (4)  and  (5). 
Prove  that  the  resulting  expressions  are  four  integrals  of  equations  (23). 

2.  Derive  equations  (23)  directly  by  taking  the  origin  at  mn,  without  first 
making  use  of  the  fixed  axes. 

3.  The  equations  (23)  are  not  symmetrical,  since  each  body  requires  a 
different  perturbative  function  ./?»,,  in  the  right    members.     Construct  the 
corresponding  system  of  differential  equations  where  the  motion  of  mn-\  is 
referred  to  a  rectangular  system  of  axes  with  the  origin  at  mn;  the  motion  of 
mn_2  to  a  parallel  system  of  axes  with  origin  at  the  center  of  mass  of  mn  and 
ran_i;  the  motion  of  mn-3  to  a  parallel  system  of  axes  with  the  origin  at  the 
center  of  mass  of  mn,  mn-\,  and  mn_2,  and  continue  in  this  way.     Show  that 
the  results  are  the  symmetrical  equations 

Mn  ffiXn-l  _      dU 

mn-l        ,.o        —  V  ,  Mn    —   Win,  Mn-1    —   Wln-l   T  ™n, 


Mn-1  fe-2  dU 

Mn-2 


-2  ~W  -  d^72 '       ^-2  =  m-2  +  mn~l  +  *"*> 


3U 

Mi=m1+m2+...  +mn, 


and  similar  equations  in  y  and  z,  where 
V  -* 


. 

Tn-l,  n-3 


(These  equations  are  the  same  as  found  by  Radau  from  a  different  standpoint 
in  the  memoir  cited  in  Art.  150.  They  have  been  employed  by  Tisserand  in 
a  very  elegant  demonstration  of  Poisson's  theorem  of  the  invariability  of  the 
major  axes  of  the  planets'  orbits  up  to  perturbations  of  the  second  order 
inclusive  with  respect  to  the  masses.  Poincare*  has  generally  used  this  system 
in  his  researches  in  the  Problem  of  Three  Bodies.) 


HISTORICAL   SKETCH.  275 

4.  Derive  the  differential  equations  corresponding  to   (23)  in  polar  co- 
ordinates. 


(J  =  1, 


HISTORICAL  SKETCH  AND   BIBLIOGRAPHY. 

The  investigations  in  the  Problem  of  n  Bodies  are  of  two  classes;  first, 
those  which  lead  to  general  theorems  holding  in  every  system;  and  second, 
those  which  give  good  approximations  for  a  certain  length  of  time  in  particular 
systems,  such  as  the  solar  system.  Investigations  of  the  second  class  are 
known  as  theories  of  perturbations,  the  discussion  of  which  will  be  given  in 
another  chapter. 

The  first  general  theorems  are  regarding  the  motion  of  the  center  of  mass, 
and  were  given  by  Newton  in  the  Principia.  The  ten  integrals  and  the 
theorems  to  which  they  lead  were  known  by  Euler.  The  next  general  result 
was  the  proof  of  the  existence  and  the  discussion  of  the  properties  of  the 
invariable  plane  by  Laplace  in  1784.  In  the  winter  semester  of  1842-43 
Jacobi  gave  a  course  of  lectures  in  the  University  of  Konigsberg  on  Dynamics. 
In  this  course  he  gave  the  results  of  some  very  important  investigations  on 
the  integration  of  the  differential  equations  which  arise  in  Mechanics.  In  all 
cases  where  the  forces  depend  upon  the  coordinates  alone,  and  where  a  po- 
tential function  exists,  conditions  which  are  fulfilled  in  the  Problem  of  n 
Bodies,  he  proved  that  if  all  the  integrals  except  two  have  been  found  the  last 
two  can  always  be  found.  He  also  showed,  in  extending  some  investigations 
of  Sir  William  Rowan  Hamilton,  that  the  problem  is  reducible  to  that  of 
solving  a  partial  differential  equation  whose  order  is  one-half  as  great  as 
that  of  the  original  system.  Jacobi's  lectures  are  published  in  the  supple- 
mentary volume  to  his  collected  works.  They  are  of  great  importance  in 
themselves,  as  well  as  being  an  absolutely  necessary  prerequisite  to  the  reading 
of  the  epoch-making  memoirs  of  Poincare,  and  they  should  be  accessible  to 
every  student  of  Celestial  Mechanics. 

It  is  a  question  of  the  highest  interest  whether  the  motions  of  the  members 
of  such  a  system  as  the  sun  and  planets  are  purely  periodic.  Newcomb  has 
shown  in  an  important  memoir  published  in  the  Smithsonian  Contributions  to 
Knowledge,  December  1874,  that  the  differential  equations  can  be  formally 
satisfied  by  purely  periodic  series.  He  did  not,  however,  prove  the  convergence 
of  these  series;  and,  indeed,  Poincare  has  shown  in  Les  Methodes  Nouvelles, 
chaps,  ix.  and  xu.,  that  they  are  in  general  divergent. 


276  HISTORICAL    SKETCH. 

As  was  stated  in  Art.  147,  Bruns  has  proved  in  the  Acta  Mathematica, 
vol.  XL,  that,  using  rectangular  coordinates,  there  are  no  new  algebraic  inte- 
grals; and  Poincare,  in  the  Acta  Mathematica,  vol.  xin.,  that,  using  the  elements 
as  variables,  there  are  no  new  uniform  transcendental  integrals,  even  when 
the  masses  of  all  the  bodies  except  one  are  very  small. 

For  further  reading  regarding  the  general  differential  equations  in  different 
sets  of  variables  the  student  will  do  well  to  consult  Tisserand's  Mecanique 
Celeste,  vol.  i.  chapters  HI.,  iv.,  and  v. 


CHAPTER  VIII. 

THE  PROBLEM  OF  THREE  BODIES. 

151.  Problem  Considered.  There  are  a  number  of  important 
results  in  the  Problem  of  Three  Bodies  which  have  been  established 
with  mathematical  rigor  if  the  initial  coordinates  and  the  com- 
ponents of  velocity  fulfill  certain  special  conditions.  While  these 
special  cases  have  not  been  found  in  nature,  there  are  nevertheless 
some  applications  of  the  results  obtained,  and  the  processes 
employed  are  mathematically  elegant  and  lead  to  most  interesting 
conclusions.  This  chapter  will  contain  such  of  these  results  as 
fall  within  the  scope  of  this  work,  reserving  the  theories  of  per- 
turbations, by  means  of  which  the  positions  of  the  heavenly  bodies 
are  predicted,  to  subsequent  chapters. 

The  first  part  of  the  chapter  will  be  devoted  to  a  discussion  of 
some  of  the  properties  of  motion  of  an  infinitesimal  body  when  it 
is  attracted  by  two  finite  bodies  which  revolve  in  circles  around 
their  center  of  mass,  and  will  include  the  proof  of  the  existence  of 
certain  particular  solutions  in  which  the  distances  of  the  infinitesi- 
mal body  from  the  finite  bodies  are  constants.  The  second  part 
of  the  chapter  will  be  devoted  to  an  exposition  of  a  method  of 
finding  particular  solutions  of  the  motion  of  three  finite  bodies  such 
that  the  ratios  of  their  mutual  distances  are  constants.  These 
solutions  include  the  former,  but  the  discoverable  properties  of 
motion  are  so  much  fewer,  and  are  obtained  with  so  much  more 
difficulty,  that  it  is  advisable  to  divide  the  discussion  into  two 
parts. 

The  particular  solutions  of  the  Problem  of  Three  Bodies  which 
will  be  discussed  here  were  given  for  the  first  time  by  Lagrange  in 
a  prize  memoir  in  1772.  The  method  adopted  here  is  radically 
different  from  that  employed  by  him,  and  lends  itself  much  more 
readily  to  a  generalization  to  the  case  where  a  larger  number  of 
bodies  is  involved.  But,  on  the  other  hand,  the  reduction  of  the 
order  of  the  problem  by  one  unit,  which  was  a  very  interesting 
feature  of  Lagrange's  memoir,  is  not  accomplished  by  this  method. 
However,  as  it  has  not  been  possible  to  make  any  use  of  this 
reduction,  it  has  not  been  of  any  practical  importance. 

Mathematically  speaking,  an  infinitesimal  body  is  one  that  is 

277 


278  THE   DIFFERENTIAL   EQUATIONS   OF  MOTION.  [152 

attracted  by  finite  masses  but  does  not  attract  them.  Physically 
speaking,  it  is  a  body  of  such  a  small  mass  that  it  will  disturb 
the  motion  of  finite  bodies  less  than  an  arbitrarily  assigned  amount, 
however  small,  during  any  arbitrarily  assigned  time,  however  long. 
To  actually  determine  a  small  mass  fulfilling  these  conditions  it  is 
only  necessary  to  make  it  so  small  that  its  whole  attraction,  which 
is  always  greater  than  its  disturbing  force,  on  one  of  the  large 
bodies,  if  placed  at  the  minimum  distance  possible,  would  move  the 
large  body  less  than  the  assigned  small  distance  in  the  assigned 
time. 

MOTION  OF  THE  INFINITESIMAL  BODY. 

152.  The  Differential  Equations  of  Motion.  Suppose  the 
system  consists  of  two  finite  bodies  revolving  in  circles  around  their 
common  center  of  mass,  and  of  an  infinitesimal  body  subject  to 
their  attraction.  Let  the  unit  of  mass  be  so  chosen  that  the  sum 
of  the  masses  of  the  finite  bodies  shall  be  unity;  then  they  can  be 
represented  by  1  —  ^  and  ju,  where  the  notation  is  so  chosen  that 
fi  ^  J.  Let  the  unit  of  distance  be  so  chosen  that  the  constant 
distance  between  the  finite  bodies  shall  be  unity.  Let  the  unit  of 
time  be  so  chosen  that  k2  shall  equal  unity.  Let  the  origin  of 
coordinates  be  taken  at  the  center  of  mass  of  the  finite  bodies, 
and  let  the  direction  of  the  axes  be  so  chosen  that  the  ^-plane  is 
the  plane  of  their  motion.  Let  the  coordinates  of  1—  n,  M>  and  the 
infinitesimal  body  be  £1,  rji,  0;  £2, 172,  0;  and  £,  rj,  f  respectively,  and 


ri  =  Vtt  -  £i)2  +  (n  -  rn)2  +  r2, 

T2  =  V(£  -  £2)2  +  (r)  -  i72)2  +  r2- 

Then  the  differential  equations  of  motion  for  the  infinitesimal 
body  are 


(1) 


i\  _       n 
=  ~ 


»y  r? 

—   *7l)  (l?    —    ^2) 


<*2r  v  r       r 

-=-(i-ti--»- 


As  a  consequence  of  the  way  the  units  have  been  chosen  the 
mean  angular  motion  of  the  finite  bodies  is 


a* 


152] 


THE   DIFFERENTIAL   EQUATIONS   OF   MOTION. 


279 


Let  the  motion  of  the  bodies  be  referred  to  a  new  system  of 
axes  having  the  same  origin  as  the  old,  and  rotating  in  the  £??- 
plane  in  the  direction  in  which  the  finite  bodies  move  with  the 
uniform  angular  velocity  unity.  The  coordinates  in  the  new 
system  are  defined  by  the  equations 


£  =  x  cos  t  —  y  sin  t, 
=  x  sin  t  +  y  cos  t, 


and  similar  equations  for  the  letters  with  subscripts  1  and  2.  On 
computing  the  second  derivatives  of  (2)  and  substituting  in  (1), 
it  is  found  that 


+  M 


(3) 


f  d?x  _  9  dy  _ 
[d?  ~       dt~ 

~{a- 

+ 


r2 


^)  cos* 


-  + 


J^!£_2-^—      1   '    /  4-  I  —  4- 
U^2          d£       ^  J  S1  1  rf^2  " 

=   -(d-M)^-^   + 


-      (1- 


+ 


sin£ 


\  cos  ^, 


df 


Multiply  the  first  two  equations  by  cos  t  and  sin  t  respectively, 
then  by  —  sin  t  and  cos  t,  and  add;  the  results  are 

(x  -  x,)          (x-  xz) 


/-72/y»  //'?/ 

_  _2— =0;—  (1  — 
rf2?/  .   ^dx 


The  position  of  the  axes  can  be  so  taken  at  the  origin  of  time 
that  the  z-axis  will  continually  pass  through  the  centers  of  the 


280 


JACOBI  S   INTEGRAL. 


[153 


finite  bodies;  then  y\  =  0,  ?/2  =  0,  and  the  equations  become 
d?x_(,dy_        _n_      ,(x-x1)_ii(x_-xz) 


(4)    H 


dzy 


dzz 


dx 


y_ 

z 
fl3 


z 
-i 


These  are  the  differential  equations  of  motion  of  the  infinitesimal 
body  referred  to  axes  rotating  so  that  the  finite  bodies  always  lie 
on  the  rr-axis.  They  have  the  important  property  that  they  do 
not  involve  explicitly  the  independent  variable  t  because  the 
coordinates  of  the  finite  bodies  have  become  constants  as  a  conse- 
quence of  the  particular  manner  in  which  the  axes  are  rotated. 
On  the  other  hand,  in  equations  (1)  the  quantities  £1,  £2,  r?i,  and  r?2 
are  functions  of  t. 

The  general  problem  of  determining  the  motion  of  the  in- 
finitesimal body  is  of  the  sixth  order;  if  it  moves  in  the  plane  of 
motion  of  the  finite  bodies,  the  problem  is  of  the  fourth  order. 

153.  Jacobi's  Integral.  Equations  (4)  admit  an  integral  which 
was  first  given  by  Jacobi  in  Comptes  Rendus  de  I' Academic  des 
Sciences  de  Paris,  vol.  in.,  p.  59,  arid  which  has  been  discussed  by 
Hill  in  the  first  of  his  celebrated  papers  on  the  Lunar  Theory, 
The  American  Journal  of  Mathematics,  vol.  i.,  p.  18,  and  again  by 
Darwin  in  his  memoir  on  Periodic  Orbits  in  Acta  Mathematica, 
vol.  xxi.,  p.  102.  Let 

a  -»:,>.. 


(5) 


U  =  i(z2  +  y2)  + 


then  equations  (4)  can  be  written  in  the  form 


(6) 


d*x         dy  _  dU 
dt*        ~di~  dx' 


dt2 


dt* 


= 
dt 


dy 


dz  ' 


If  these    equations  are    multiplied  by  2  —  ,  2  -^ ,  and  2  -r   re- 
spectively, and  added,  the  resulting  equation  can  be  integrated, 


154]  THE    SURFACES    OF   ZERO    RELATIVE    VELOCITY.  281 

since  U  is  a  function  of  x,  y,  and  z  alone,  and  give 


Five  integrals  more  are  required  in  order  completely  to  solve 
the  problem.  If  the  infinitesimal  body  moved  in  the  xy-plane 
only  three  would  remain  to  be  found,  the  last  two  of  which  could 
be  obtained  by  Jacobi's  last  multiplier,*  if  the  first  one  were  found. 
Thus  it  appears  that  only  one  new  integral  is  needed  for  the  com- 
plete solution  of  this  special  problem  in  the  plane.  f  But  Bruns 
has  proved  in  Ada  Mathematica,  vol.  xi.,  that,  when  rectangular 
coordinates  are  used,  no  new  algebraic  integrals  exist;  and  Poin- 
care  has  proved  in  Les  Methodes  Nouvelles  de  la  Mecanique  Celeste, 
vol.  i.,  chap,  v.,  that  when  the  elements  of  the  orbits  are  used  as 
variables,  there  are  no  new  uniform  transcendental  integrals, 
even  when  the  mass  of  one  of  the  finite  bodies  is  very  small  com- 
pared to  that  of  the  other  (see  Art.  147).  These  demonstrations 
are  entirely  outside  the  scope  of  this  work  and  cannot  be  repro- 
duced here. 

154.  The  Surfaces  of  Zero  Relative  Velocity.  J  Equation  (7) 
is  a  relation  between  the  square  of  the  velocity  and  the  coordinates 
of  the  infinitesimal  body  referred  to  the  rotating  axes.  Therefore, 
when  the  constant  of  integration  C  has  been  determined  numeri- 
cally by  the  initial  conditions,  equation  (7)  determines  the  velocity 
with  which  the  infinitesimal  body  will  move,  if  at  all,  at  all.  points 
of  the  rotating  space;  and  conversely,  for  a  given  velocity,  equa- 
tion (7)  gives  the  locus  of  those  points  of  relative  space  where  alone 
the  infinitesimal  body  can  be.  In  particular,  if  V  is  put  equal  to 
zero  in  this  equation  it  will  define  the  surfaces  at  which  the  velocity 
will  be  zero.  On  one  side  of  these  surfaces  the  velocity  will  be 
real  and  on  the  other  side  imaginary;  or,  in  other  words,  it  is 

*  Developed  in  Vorlesungen  uber  Dynamik,  supplementary  volume  to 
Jacobi's  collected  works. 

t  Hill  put  his  special  equations  in  such  a  form  that  they  would  be  reduced 
to  quadratures  if  a  single  variable  were  expressed  in  terms  of  the  time,  American 
Journal  of  Mathematics,  vol.  i.,  p.  16. 

t  First  discussed  by  Hill  in  his  Lunar  Theory,  The  American  Journal  of 
Mathematics,  vol.  i.;  and  again,  for  motion  in  the  xy-plane,  by  Darwin  in  his 
Periodic  Orbits,  in  Ada  Mathematica,  vol.  xxi. 


282  APPROXIMATE    FORMS    OF   THE    SURFACES.  [155 

possible  for  the  body  to  move  on  one  side,  and  impossible  for  it 
to  move  on  the  other.  The  general  proposition  that  a  function 
changes  sign  as  the  surface  at  which  it  is  zero  is  crossed  (at  least 
at  a  regular  point  of  the  surface)  was  proved  in  Art.  120.  While 
it  will  not  be  possible  to  say  in  any  except  very  particular  cases 
what  the  orbit  will  be,  yet  this  partition  of  relative  space  will 
show  in  what  portions  the  infinitesimal  body  can  move  and  in 
what  portions  it  can  not. 

The  equation  of  the  surfaces  of  zero  relative  velocity  is 


(8) 


+  y2  +  z2, 


I  r2  =   4(x  -  z2)2  +  y2  +  z2. 


Since  only  the  squares  of  y  and  z  occur  the  surfaces  defined  by  (8) 
are  symmetrical  with  respect  to  the  xy  and  zz-planes,  and,  when 
/A  =  |,  with  respect  to  the  j/z-plane  also.  The  surfaces  for  JJL  =j=  J 
can  be  regarded  as  being  deformations  of  those  for  /*  =  J.  It 
follows  from  the  way  in  which  z  enters  that  a  line  parallel  to  the 
z-axis  pierces  the  surfaces  in  two  (or  no)  real  points.  Moreover, 
the  surfaces  are  contained  within  a  cylinder  whose  axis  is  the 
z-axis  and  whose  radius  is  VC,  to  which  certain  of  the  folds  are 
asymptotic  at  z2  =  °o  ;  for,  as  z2  increases  the  equation  approaches 
as  a  limit 

z2  +  y2  =  C. 

155.  Approximate  Forms  of  the  Surfaces.  From  the  properties 
of  the  surfaces  given  in  the  preceding  article  and  from  the  shapes 
of  the  curves  in  which  the  surfaces  intersect  the  reference  planes, 
a  general  idea  of  their  form  can  be  obtained.  The  equation  of 
the  curves  of  intersection  of  the  surfaces  with  the  xy-pl&ne  is 
obtained  by  putting  z  equal  to  zero  in  the  first  of  (8),  and  is 


(x-xtf  +  y*      A(z  -  *2)2  +  2/2 

For  large  values  of  x  and  y  which  satisfy  this  equation  the  third 
and  fourth  terms  are  relatively  unimportant,  and  the  equation 
may  be  written 


155] 


APPROXIMATE  FORMS  OF  THE  SURFACES. 


283 


where  e  is  a  small  quantity.  This  is  the  equation  of  a  circle  whose 
radius  is  A/C  —  e;  therefore,  one  branch  of  the  curve  in  the  in/- 
plane is  an  approximately  circular  oval  within  the  asymptotic 
cylinder.  It  is  also  to  be  noted  that  the  larger  C  is,  the  larger 
are  the  values  of  x  and  y  which  satisfy  the  equation,  the  smaller 
is  e,  the  more  nearly  circular  is  the  curve,  and  the  more  nearly 
does  it  approach  its  asymptotic  cylinder. 


x-Aocis 


Fig.  38. 

For  small  values  of  x  and  y  satisfying  (9)  the  first  and  second 
terms  are  relatively  unimportant,  and  the  equation  may  be 
written 


1  —  M    ,    M 


This  is  the  equation  of  the  equipotential  curves*  for  the  two  centers 
of  force,  1  —  n  and  n>  For  large  values  of  C  they  consist  of 
closed  ovals  around  each  of  the~bodies  1  —  M  and  M;  for  smaller 
values  of  C  these  ovals  unite  between  the  bodies  forming  a  dumb- 

*  Thomson  and  Tait's  Natural  Philosophy,  Part  II.,  Art.  508. 


284 


APPROXIMATE    FORMS    OF   THE    SURFACES. 


[155 


bell  shaped  figure  in  which  the  ends  are  of  different  size  except 
when  n  =  J;  and  for  still  smaller  values  of  C  the  handle  of  the 
dumb-bell  enlarges  until  the  figure  becomes  an  oval  enclosing 
both  of  the  bodies 

From  the  foregoing  considerations  it  follows  that  the  approxi- 
mate forms  of  the  curves  in  which  the  surfaces  intersect  the  in/- 
plane are  as  given  in  Fig.  38.  The  curves  Ci,  C2,  Cs,  C4,  CB  are 
in  the  order  of  decreasing  values  of  the  constant  C.  They  were 
not  drawn  from  numerical  calculations  and  are  intended  to  show 
only  qualitatively  the  character  of  the  curves. 


z  -Axis 


Fig.  39. 

The  equation  of  the  curves  of  intersection  of  the  surfaces  and 
the  rcz-plane  is  obtained  by  putting  y  equal  to  zero  in  equation 
(8),  and  is 

2(1  -  M)  2M 


(10) 


=  C. 


For  large  values  of  x  and  z  satisfying  this  equation  the  second 


155] 


APPROXIMATE  FORMS  OF  THE  SURFACES. 


285 


and  third  terms  are  relatively  unimportant,  and  it  may  be  written 

x2  =  C  -  e, 

which  is  the  equation  of  a  symmetrical  pair  of  straight  lines 
parallel  to  the  z-axis.  The  larger  C  is,  the  larger  is  the  value  of  x 
which,  for  a  given  value  of  z,  satisfies  the  equation,  and,  therefore, 
the  smaller  is  e.  Hence,  the  larger  C  the  closer  the  lines  are  to  the 
asymptotic  cylinder. 

z  -Axis 


y—Axis 


Fig.  40. 

For  small  values  of  x  and  z  satisfying  equation  (10)  the  first 
term  is  relatively  unimportant,  and  the  equation  may  be  written 


fl    =C 

r2       2 


6. 


This  is  again  the  equation  of  the  equipotential  curves  and  has  the 
same  properties  as  before.  Hence,  the  forms  of  the  curves  in  the 
zz-plane  are  qualitatively  like  those  given  in  Fig.  39.  Again, 
the  curves  Ci,  •  •  • ,  CB  are  in  the  order  of  decreasing  values  of  the 
constant  C,  and  were  not  drawn  from  numerical  calculations. 
The  equation  of  the  curves  of  intersection  of  the  surfaces  and 


286  THE   REGIONS   OF   REAL   AND   IMAGINARY   VELOCITY.  [156 

the  2/z-plane  is  obtained  by  putting  x  equal  to  zero  in  equation 
(8),  and  is 
(11)        y.  2(1  -M)  2M 


> 

+  I/2  +  22          VZ22  +  ?/2  +  Z2 

For  large  values  of  y  and  2  satisfying  this  equation  the  second  and 
third  terms  are  relatively  unimportant,  and  it  may  be  written 

2/2  =  C  -  e, 

which  is  the  equation  of  a  pair  of  lines  near  the  asymptotic  cylinder, 
approaching  it  as  C  increases. 

If  1  —  ju  is  much  greater  than  jj,,  the  numerical  value  of  #2  is 
much  greater  than  that  of  x\\  hence,  for  small  values  of  y  and  z 
satisfying  (11),  this  equation  may  be  written 


rl 

which  is  the  equation  of  a  circle  which  becomes  larger  as  C  de- 
creases. Hence,  the  forms  of  the  curves  in  the  t/2-plane  are  quali- 
tatively as  given  in  Fig.  40.  Again,  the  curves  Ci,  •  •  • ,  CB  are 
in  the  order  of  decreasing  values  of  the  constant  C. 

From  these  three  sections  of  the  surfaces  it  is  easy  to  infer  their 
forms  for  the  different  values  of  C.  They  may  be  roughly  de- 
scribed as  consisting  of,  for  large  values  of  C,  a  closed  fold  approxi- 
mately spherical  in  form  around  each  of  the  finite  bodies,  and  of 
curtains  hanging  from  the  asymptotic  cylinder  symmetrically 
with  respect  to  the  xy-pl&ue;  for  smaller  values  of  C,  the  folds 
expand  and  coalesce  (Fig.  38,  curve  C3);  for  still  smaller  values 
of  C  the  united  folds  coalesce  with  the  curtains,  the  first  points  of 
contact  being  in  every  case  in  the  :n/-plane;  and  for  sufficiently 
small  values  of  C  the  surfaces  consist  of  two  parts  symmetrical 
with  respect  to  the  :n/-plane  but  not  intersecting  it  (Figs.  39, 
curve  C&,  and  40,  curve  Ce). 

156.  The  Regions  of  Real  and  Imaginary  Velocity.  Having 
determined  the  forms  of  the  surfaces,  it  remains  to  find  in  what 
regions  of  relative  space  the  motion  is  real  and  in  what  it  is  imagi- 
nary. The  equation  for  the  square  of  the  velocity  is 

e\/-t  \  r» 


Suppose  C  is  so  large  that  the  ovals  and  curtains  are  all  separate. 


157]  METHOD   OF    COMPUTING   THE   SURFACES.  287 

The  motion  will  be  real  in  those  portions  of  relative  space  for 
which  the  right  member  of  this  equation  is  positive.  If  it  is 
positive  in  one  point  in  a  closed  fold  it  will  be  positive  in  every 
other  point  within  it,  for  the  function  changes  sign  only  at  a  surface 
of  zero  relative  velocity. 

It  is  evident  from  the  equation  that  x  and  y  can  be  taken  so 
large  that  the  right  member  will  be  positive,  however  great  C  may 
be;  therefore,  the  motion  is  real  outside  of  the  curtains.  It  is  also 
clear  that  a  point  can  be  chosen  so  near  to  either  1  —  JJL  or  /*,  that 
is,  either  ri  or  r2  may  be  taken  so  small,  that  the  right  member  will 
be  positive,  however  great  C  may  be;  therefore,  the  motion  is  real 
within  the  folds  around  the  finite  bodies. 

If  the  value  of  C  were  so  large  that  the  folds  around  the  finite 
bodies  were  closed,  and  if  the  infinitesimal  body  should  be  within 
one  of  these  folds  at  the  origin  of  time,  it  would  always  remain 
there  since  it  could  not  cross  a  surface  of  zero  velocity.  If  the 
earth's  orbit  is  supposed  to  be  circular  and  the  mass  of  the  moon 
infinitesimal,  it  is  found  that  the  constant  C,  determined  by  the 
motion  of  the  moon,  is  so  large  that  the  fold  around  the  earth  is 
closed  with  the  moon  within  it.  Therefore  the  moon  cannot 
recede  indefinitely  from  the  earth.  It  was  in  this  manner,  and 
with  these  approximations,  that  Hill  proved  that  the  moon's 
distance  from  the  earth  has  a  superior  limit.* 

157.  Method  of  Computing  the  Surfaces.  Actual  points  on 
the  surfaces  can  be  found  most  readily  by  first  determining  the 
curves  in  the  :n/-plane,  and  then  finding  by  methods  of  approxi- 
mation the  values  of  z  which  satisfy  (7).  Besides,  the  curves  in 
the  rn/-plane  are  of  most  interest  because  the  first  points  of  contact 
as  the  various  folds  coalesce  occur  in  this  plane,  and,  indeed,  on 
the  x-axis,  as  can  be  seen  from  the  symmetries  of  the  surfaces. 

The  equation  of  the  curves  in  the  xy-plane  is 


\  tf  \  "  M         \ 


z  -  *i        y          x  -  *2        y 

If  this  equation  is  rationalized  and  cleared  of  fractions  the  result 
is  a  polynomial  of  the  sixteenth  degree  in  x  and  y.  When  the  value 
of  one  of  the  variables  is  taken  arbitrarily  the  corresponding 
values  of  the  other  can  be  found  by  solving  this  rationalized 
equation.  This  problem  presents  great  practical  difficulties 
*  Lunar  Theory,  Am.  Jour.  Math.,  vol.  i.,  p.  23. 


288  METHOD    OF    COMPUTING   THE    SURFACES.  [157 

because  of  the  high  degree  of  the  equation,  and  these  troubles 
are  supplemented  by  the  presence  of  foreign  solutions  which  are 
introduced  by  the  processes  of  rationalization. 

The  difficulty  from  foreign  solutions  can  be  avoided  entirely, 
and  the  degree  of  the  equation  can  be  very  much  reduced  by 
transforming  to  bi-polar  coordinates.  That  is,  points  on  the 
curves  can  be  denned  by  giving  their  distances  from  two  fixed 
points  on  the  o>axis.  This  method  could  not  be  applied  if  the 
curves  were  not  symmetrical  with  respect  to  the  axis  on  which 
the  poles  lie.  Let  the  centers  of  the  bodies  1  —  M  and  /*  be  taken 
as  the  poles;  the  distances  from  these  points  are  r\  and  r2  respec- 
tively. To  complete  the  transformation  it  is  only  necessary  to 
express  x2  +  y2  in  terms  of  these  quantities. 


y -\axia 


X-axis 
-4 

Fig.  41. 


Let  P  be  a  point  on  one  of  the  curves;  then  OA  =_x,  AP  =  y, 
and,  since  0  is  the  center  of  mass  of  1  —  /z  and  /*,  OM  =  1  —  ju, 
and  0(1  -  /z)  =  -  M-  It  follows  that 


r  jf    =   fli   _    (X  +  M)2 
[  =   r,2   _     x   _      1    _ 


=   ^2   _   X2  +  2(1    -   rfx   ~    (I    ~   M)2- 

On  eliminating  the  first  power  of  x  from  these  equations  and  solv- 
ing for  x2  +  y2,  it  is  found  that 

x2  +  y2  =  (1  -  ju)n2  +  jur22  -  /*(!  -  /*)• 
As  a  consequence  of  this  equation,  (9)  becomes 

(12)      (1  -  M)     Tl2  +         +  M    r22  +         =  C  +  /il-/=C". 


If  an  arbitrary  value  of  r2  is  assumed  n  can  be  computed  from 
this  equation;  the  points  of  intersection  of  the  circles  around 
1  —  ;u  and  v  as  centers,  with  the  computed  and  assumed  values 
respectively  of  r\  and  r2  as  radii,  will  be  points  on  the  curves.  To 
follow  out  this  plan,  let  equation  (12)  be  written  in  the  form 


157] 


METHOD    OF    COMPUTING   THE    SURFACES. 


289 


0, 


(13) 


=  2. 


Since  b  =  2  is  positive  there  is  at  least  one  real  negative  root  of 
the  first  of  (13)  whatever  value  a  may  have.  But  the  only  value 
of  n  which  has  a  meaning  in  this  problem  is  real  and  positive; 
hence  the  condition  for  real  positive  roots  must  be  considered. 

It  follows  from  (12)  that  C"  is  always  greater  than  /*    r22  H  — 

for  all  real  positive  values  of  r\  and  r2  ;  therefore  a  is  always  nega- 
tive. It  is  shown  in  the  Theory  of  Equations  that  a  cubic  equa- 
tion of  this  form  has  three  distinct  real  roots  if  2762  +  4a3  <  0; 
or,  since  b  =  2,  if 

(14)  a  +  3  <  0. 


Suppose  this  inequality  is  satisfied. 
of  solving  the  cubic  is 


Then  a  convenient  method 


(15) 


where  rn,  r*i2,  rn  are  the  three  roots  of  the  cubic. 

The  limit  of  the  inequality  (14)  is  a  +  3  =  0;  or,  in  terms  of 
the  original  quantities, 


(16) 


r23  +  aV2  +  &'  =  0, 


b'  =  2. 


The  solution  of  this  equation  gives  the  extreme  values  of  r2  for 
which  (13)  has  real  roots.  Therefore,  in  the  actual  computation 
equation  (16)  -should  be  solved  first  for  r2i  and  r22.  The  values  of 

20 


290  PARTICULAR   SOLUTIONS    OF  [158 

r2  to  be  substituted  in  (13)  should  be  chosen  at  convenient  inter- 
vals between  these  roots. 

Equation  (16)  will  not  have  real  positive  roots  for  all  values 
of  a',  the  condition  for  real  positive  roots  being 

a'  +  3  ^  0; 

the  limiting  value  of  which  is,  in  the  original  quantities, 
C"      3(1-,.) 

--  T  '  ---  —     —    O, 

M  M 

whence 

C'  =  3. 

Therefore  C'  must  be  equal  to,  or  greater  than,  3  in  order  that  the 
curves  shall  have  real  points  in  the  xy-pl&ne.  For  C'  =  3  the 
curves  are  just  vanishing  from  the  plane,  and  it  follows  at  once 
\  that  equation  (12)  is  then  satisfied  by  r\  =  1,  r2  =  1;  that  is,  the 
surfaces  vanish  from  the  xy-pl&ne  at  the  points  which  form  equi- 
lateral triangles  with  1  —  M  and  M- 

158.  Double  Points  of  the  Surfaces  and  Particular  Solutions 
of  the  Problem  of  Three  Bodies.  It  follows  from  the  general 
forms  of  the  surfaces  that  the  double  points  which  appear  as  C 
diminishes  are  all  in  the  rri/-plane.  Therefore  it  is  sufficient  in 
this  discussion  to  consider  the  equation  of  the  curves  in  the 
zi/-plane.  There  are  three  double  points  on  the  z-axis  which 
appear  when  the  ovals  around  the  finite  bodies  touch  each  other 
and  when  they  touch  the  exterior  curve  enclosing  them  both. 
There  are  two  more  which  appear,  as  the  surfaces  vanish  from  the 
zi/-plane,  at  the  two  points  making  equilateral  triangles  with  the 
finite  bodies.  • 

These  double  points  are  of  interest  as  critical  points  of  the 
curves,  and  it  will  now  be  shown  that  they  are  connected  with 
important  dynamical  properties  of  the  system.  Let  the  equation 
of  the  curves  be  written 


The  conditions  for  double  points  are 


(17)      F(x,  y)ma»  +  1f 

ouble    oints  are 


1  dF  .          ,  (x  —  Xi)          (x  —  Xz)       n 

o  T~  =  x  —   (1   —  M) 5—^  —  M  3 5 i  =  0; 

2  dz  rx3  r23 

A  " 

la^7  «         ?/ 

2  dy  ~  y  ^  ri3       M  r23  * 


158]  THE   PROBLEM    OF   THREE    BODIES.  291 

The  left  members  of  these  equations  are  the  same  as  the  right 

1  r^ff 

members  of  the  equations  (4)  for  z  =  0.     The  expressions  -  - 

Z  ox 

1      r)  W 

and  -  —  are  proportional  to  the  direction  cosines  of  the  normal 

at  all  ordinary  points  of  the  curves;  and  since  3-  and  -jr  are  zero 

at          at 

at  the  surfaces  of  zero  velocity  it  follows  from  (4)  that  the  directions 
of  acceleration,  or  the  lines  of  effective  force,  are  orthogonal  to  the 
surfaces  of  zero  relative  velocity.  Therefore,  if  the  infinitesimal 
body  is  placed  on  a  surface  of  zero  relative  velocity  it  will  start 
in  its  motion  in  the  direction  of  the  normal.  But  at  the  double 
points  the  sense  of  the  normal  becomes  ambiguous;  hence,  it  might 
be  surmised  that  if  the  infinitesimal  body  were  placed  at  one  of 
these  points  it  would  remain  relatively  at  rest. 

The  conditions  imposed  by  (17)  and  (18)  are  also  the  conditions 

that  -JTJ  and  -^  ,  or  the  components  of  acceleration,  in  equations 

(4)  shall  vanish.  Hence,  if  the  infinitesimal  body  is  placed  at  a 
double  point  with  zero  relative  velocity,  its  coordinates  will  identically 
fulfill  the  differential  equations  of  motion  and  it  will  remain  forever 
relatively  at  rest,  unless  disturbed  by  forces  exterior  to  the  system 
under  consideration.  These  are  particular  solutions  of  the  Problem 
of  Three  Bodies,  and  are  special  cases  of  the  Lagrangian  solutions. 
Consider  equations  (18),  the  second  of  which  is  satisfied  by 
y  =  0.  The  double  points  on  the  z-axis,  and  the  straight  line 
solutions  of  the  problem  are  given  by  the  conditions 

(x  -  xi)  (x  .-  xt) 


(19) 


-- 

y  =  o, 

z  =  0. 


The  left  member  of  the  first  equation  considered  as  a  function 
of  x  is  positive  f or  x  =  +  oo  ;  it  is  negative  for  x  =  x2  +  e,  where  e 
is  a  very  small  positive  quantity;  it  is  positive  for  x  =  £2  —  e; 
it  is  negative  for  x  =  x\  +  e;  it  is  positive  for  x  =  Xi  —  e;  and  it 
is  negative  for  x  =  —  oo.  Since  the  function  is  finite  and  con- 
tinuous except  when  x  =  +  <*> ,  x*,  x\,  or  --  oo,  it  follows  that 
the  function  changes  sign  three  times  by  passing  through  zero, 
(a)  once  between  +  oo  and  xz,  (b)  once  between  z2  and  Xi,  and 
(c)  once  between  xi  and  —  oo.  Therefore,  there  are  three  posi- 


292  PARTICULAR   SOLUTIONS   OF  [158 

tions  on  the  line  through  1  —  /*  and  M  at  which  the  infinitesimal 
body  will  remain  when  given  proper  initial  projection. 

(a)  Let  the  distance  from  ^  to  the  double  point  on  the  #-axis 
between  +  oo  and  x2  be  represented  by  p.  Then  x  —  #2  =  P, 
x  —  Xi  =  ri  =  1  +  p,  x  =  I  —  M  +  P;  therefore  the  first  equation 
of  (19)  becomes  after  clearing  of  fractions 

(20)     p6  +  (3  -  /x)p4  +  (3  -  2/z)p3  -  jup2  -  2/zp  -  M  =  0. 

This  quintic  equation  has  one  variation  in  the  sign  of  its  coef- 
ficients, and  hence  only  one  real  positive  root.  The  value  of  this 
root  depends  upon  /*.  Consider  the  left  member  of  the  equation 
as  a  function  of  p  and  ju.  For  /*  =  0  the  equation  becomes 

PV  +  3P  +  3)  =  0, 

which  has  three  roots  p  =  0,  and  two  others,  coming  from  the 
second  factor,  which  are  complex.  It  follows  from  the  theory 
of  the  solution  of  algebraic  equations  that,  for  /JL  different  from 
zero  but  sufficiently  small,  three  roots  of  the  equation  are  ex- 
pressible as  power  series  in  /**,  vanishing  with  this  parameter.* 
The  one  of  these  three  roots  obtained  by  taking  the  real  value  of  /** 
is  real;  the  other  two  are  complex.  Therefore,  the  real  root  has 
the  form 

On  substituting  this  expression  for  p  in  (20)  and  equating  to  zero 
the  coefficients  of  corresponding  powers  of  M*>  it  is  found  that 

_  3*  _  3*  1 

ai~3~'         a2~~9'  ~27' 

Hence 

(21) 

P- 

The  corresponding  value  of  C"  is  found  by  substituting  these 
values  of  r\  and  r2  in  equation  (12). 

(6)  Let  the  distance  from  /*  to  the  double  point  on  the  x- 
axis  between  x%  and  x\  be  represented  by  p.  Then  in  this  case 
x  —  x2  =  —  p,  x  —  x\  =  r\  =  1  —  p,  x  =  (1  —  M)  —  P;  therefore 
the  first  equation  of  (19)  becomes 

p5  -  (3  -  M)p4  +  (3  -  2^)p3  -  MP2  +  2/zp  -  M  =  0. 
*  See  Harkness  and  Morley's  Theory  of  Functions,  chapter  iv. 


158]  THE    PROBLEM    OF   THREE    BODIES.  293 

On  solving  as  in  (a),  the  values  of  r2  and  r\  are  found  to  be 
r  /M\* 

(22)    r2  =  p  =  UJ  "  . 

In';-  i-p- 

The  corresponding  value  of  C'  is  found  by  substituting  these 
values  of  r\  and  r2  in  equation  (12). 

(c)  Let  the  distance  from  1  —  /*  to  the  double  point  on  the 
z-axis  between  x\  and  —  oo  be  represented  by  1  —  p.  In  this  case 
z  -  z2  =  -  2  +  p,  z  -  Zi  =  -  1  +  p,  x=-fj.-l  +  p,  and 
the  first  equation  of  (19)  becomes 

P6  -  (7  +  M)P4  +  (19  +  6M)p3  -  (24  +  13M)p2 

( ^o ) 

+  (12  +  14M)P  -  7M  =  0. 
When  /*  =  0  this  equation  becomes 

P8  -  7p4  +  19p3  -  24P2  +  12p  =  0, 

which  has  but  one  root  p  =  0.  Therefore  p  can  be  expressed  as  a 
power  series  in  /*  which  converges  for  sufficiently  small  values  of 
this  parameter,  and  vanishes  with  it.  This  root  will  have  the 
form 

P  =  Cin  +  c2ju2  +  c3M3  +  c4M4  +•••'•• 

On  substituting  this  expression  for  p  in  (23),  and  equating  to  zero 
the  coefficients  of  the  various  powers  of  M,  it  is  found  that 

7  23  X  72 

Cl=l2'         °2  =    '         Cs  =    T24 — ' 
Hence 

7       ,  23  X  72  3   . 
P  =  12M+      124     M+    B| 

(24) 

1      =  1  -  p, 

2  =  1  +  ri  =  2  -  p. 

The  corresponding  value  of  C'  is  found  by  substituting  these 
values  of  r\  and  r2  in  equation  (12). 

If  the  values  of  r\  and  r2  given  by  the  first  three  terms  of  the 
series  (21),  (22),  and  (24)  are  not  sufficiently  accurate,  more 
nearly  correct  values  should  be  found  by  differential  corrections. 

In  order  to  find  the  double  points  not  on  the  z-axis  consider 
equations  (18)  again.  They,  or  any  two  independent  functions 
of  them,  define  the  double  points.  Since  y  is  distinct  from  zero 
in  this  case  the  second  equation  may  be  divided  by  it,  giving 


294 


PROBLEMS. 


1  _ 


_  JL  =  o. 


Multiply  this  equation  by  x  —  rc2,  and  x  —  xi}  and  subtract  the 
products  separately  from  the  first  of  (18).     The  results  are 


But   x2  —  1  —  ju>   X 
equations  reduce  to 


=  —  ^    and   £2  —  £1  =  1;   therefore   these 


-  1  +  -,  \  =  0, 
r23 

2  =  0. 

The  only  real  solutions  are  r\  =  1,  r2  =  1,  and  the  points  form 
equilateral  triangles  with  the  finite  bodies  whatever  their  relative 
masses  may  be.  As  was  shown  in  the  last  of  Art.  157,  they  occur 
at  the  places  where  the  surfaces  vanish  from  the  xy-plane. 


XX.     PROBLEMS. 

1.  The  units  defined  in  Art.  152  are  called  canonical  units;  what  would 
the  canonical  unit  of  time  be  in  days  for  the  earth  and  sun? 

2.  Show  on  d  priori  grounds  that,  when  the  niotion  of  the  system  is  referred 
to  axes  rotating  as  in  Art.  152,  the  differential  equations  should  not  involve 
the  time  explicitly. 

3.  Why  cannot  an  integral  corresponding  to  (7)  be  derived  from  equations 
(1)  at  once  without  any  transformations?     Prove  that  there  is  an  integral 
of  (1). 

4.  What  are  the  surfaces  of  zero  velocity  for  a  body  projected  vertically 
upward  against  gravity?     For  a  body  moving  subject  to  a  central  force 
varying  inversely  as  the  square  of  the  distance? 

5.  Show  by  direct  reductions  from  (13)  and  (14)  that 


—  rn)(ri  —  ri2)(n  -  r13) 


+ 


+  6  =  0. 


6.  Prove  that  the  solution  of  (16)  gives  the  extreme  values  of  r2  for  which 
(14)  has  real  roots,         Hint.     Consider  the  graph  of  y  =  r23  +  a'r2  +  b'. 


159]       TISSERAND'S  CRITERION  FOR  IDENTITY  OF  COMETS.       295 

7.  Impose  the  conditions  on  (12)  that  C"  shall  be  a  minimum  and  show 
that  it  is  satisfied  only  for  rL  =  1,  r2  =  1,  and  that  the  minimum  value  of  C' 
is  3. 

8.  Why  are  not  the  lines  of  effective  force  orthogonal  to  all  of  the  surfaces 
of  constant  velocity? 

9.  Prove  that  the  double  point  between  /j,  and  1  —  ^  is  nearer  /*  than  is 
the  one  between  ju  and  +  « . 

10.  Prove  that,  as  C'  diminishes,  the  first  double  point  to  appear  is  the  one 
between  /j.  and  1  —  /*;  the  second,  the  one  between  p.  and  +  °°  J  the  third, 
the  one  between  I  —  n  and  —  w  ;  and  the  last,  those  which  make  equilateral 
triangles  with  the  finite  bodies. 

11.  If  /i  =  TT>  1  -  M  =  TT,  find  the  values  of  n,  r2,  and  C'  from  (21),  (22), 
(24),  and  (12). 

1(21)  r2  =  0.340,  ri  =  1.340,  C'  =  3.535; 
(22)  r2  =  0.276,  n  =  0.724,  C'  =  3.653; 
(24)  r2  =  1.947,  ri  =  0.947,  C"  =  3.173. 

12.  From  the  approximate  values  of  the  last  example  find  by  the  method 
of  differential  corrections  more  accurate  values. 

f  (21)     r2  =  0.347,        n  =  1.347,        C'  =  3.534; 

Ans.    J  (22)    r2  =  0.282,        n  =  0.718,        C'  =  3.653; 

[  (23)     r2  =  1.947,        n  =  0.947,        C'  =  3.173. 

13.  Considering  the  earth's  orbit  to  be  a  circle,  find  the  distance  in  miles 
from  the  earth  to  the  double  point  which  is  opposite  to  the  sun.     Would  an 
infinitesimal  body  at  this  point  be  eclipsed? 

Ans.     930,240  miles. 

159.  Tisserand's  Criterion  for  the  Identity  of  Comets.*  Comets 
sometimes  pass  near  the  planets  in  their  revolutions  around  the 
sun,  and  then  the  elements  of  their  orbits  are  greatly  changed. 
The  planet  Jupiter  is  especially  potent  in  producing  these  per- 
turbations because  of  its  great  mass  and  because  at  its  distance 
the  attraction  of  the  sun  is  much  less  than  it  is  at  the  distances  of 
the  earth-like  planets.  Since  a  comet  has  no  characteristic 
features  by  which  it  may  be  recognized  with  certainty,  its  identity 
might  be  in  question  if  it  were  not  followed  visually  during  the 
time  of  the  perturbations. 

One  way  of  testing  the  identity  of  two  comets  appearing  at 
different  epochs  is  to  take  the  orbit  of  the  earlier  and  to  compute 
the  perturbations  which  it  undergoes,  and  then  to  compare  the 
derived  elements  with  those  determined  from  the  later  obser- 

*  Bulletin  Astronomique,  vol.  vi.,  p.  289,  and  Mec.  Cel.,  vol.  iv.,  p.  203. 


296       TISSERAND'S  CRITERION  FOR  IDENTITY  OF  COMETS.       [159 

vations;  or,  the  start  may  be  made  with  the  elements  of  the  later 
comet,  and  by  inverse  processes  the  earlier  elements  may  be  com- 
puted and  the  comparison  made.  One  or  the  other  of  these  plans 
has  been  followed  until  recent  years. 

But  the  question  arises  if  there  is  not  some  relation  among  the 
elements  which  remains  unaltered  by  the  perturbations.  This 
is  the  question  which  Tisserand  has  answered  in  the  affirmative  in 
one  of  his  characteristically  elegant  and  important  papers  on 
Celestial  Mechanics. 

Let  the  eccentricity  of  Jupiter's  orbit  be  supposed  equal  to  zero, 
and  the  mass  of  the  comet  infinitesimal.  While  both  of  these 
assumptions  are  false  they  are  very  nearly  fulfilled,  and  the  error 
introduced  will  be  inappreciable,  especially  as  the  comet  will  be 
near  enough  to  Jupiter  to  suffer  sensible  disturbances  only  a  very 
short  time.  Under  these  suppositions,  and  when  the  units  are 
properly  chosen,  the  integral 


holds  true.  This  is  an  answer  to  the  question;  for,  when  the 
elements  are  known  the  velocity  and  coordinates  can  be  computed 
at  any  time,  and  the  motion  referred  to  rotating  axes  by  equations 
(2).  Hence,  to  test  the  identity  of  two  comets,  compute  the 
function  (7)  for  each  orbit  and  see  if  the  constant  C  is  the  same 
for  both.  If  the  two  values  of  C  are  the  same,  the  probability  is 
very  strong  that  only  one  comet  has  been  observed;  if  they  are 
different,  the  two  comets  are  certainly  distinct  bodies. 

The  process  just  explained  has  the  inconvenience  of  involving 
considerable  computati9n.  This  can  be  largely  avoided  by  ex- 
pressing (7)  in  terms  of  the  ordinary  elements  of  the  orbit.  The 
first  step  is  to  express  (7)  in  terms  of  coordinates  measured  from 
fixed  axes.  The  equations  of  transformation  are  the  inverse  of 
equations  (2),  viz., 

'  x  =  +  %  cos  t  +  17  sin  t, 
y  =  —  %  sin  t  +  77  cos  t, 
z  =  f. 

From  these  equations  it  is  found  that 


159]       TISSERAND'S  CRITERION  FOR  IDENTITY  OF  COMETS.        297 


Hence  equation  (7)  becomes 


=  2U  ~  M)    ,  2^  _  ^ 
T\  TZ 

Let  r  represent  the  distance  of  the  comet  from  the  origin,  and  i 
the  angle  between  the  plane  of  its  instantaneous  orbit  and  the 
£i7-plane.  Then  equations  (24),  Art.  89,  give 


dt  \dt  /        \dt 


Hence  equation  (25)  becomes 
(26)     ?_i_ 


ri  r2 

In  the  case  of  Jupiter  and  the  sun  ju  is  less  than  one-thousandth. 
Therefore  the  origin  is  very  near  the  center  of  the  sun,  and  TI  is 
sensibly  equal  to  r.  In  both  instances  the  elements  will  be  deter- 
mined when  the  comet  is  far  from  both  Jupiter  and  the  sun  so  that 

2u        2u 
--  -  H  —  -  will   be  so  small  that  it  may  be  neglected  without 

fl         Tz  oj 

important  error;  then  (26)  reduces  to  the  simple  expression 

cos  i  =  C. 

It  will  be  noticed  that  the  elements  of  this  formula  are  the 
instantaneous  elements  for  motion  around  a  unit  mass  situated 
at  the  center  of  mass  of  the  finite  bodies.  The  actual  elements 
used  in  Astronomy  are  the  elements  referred  to  the  center  of  the 
sun,  with  the  sun  as  the  attracting  mass.  Nevertheless,  on 
account  of  the  small  relative  mass  of  Jupiter  the  two  sets  of 
elements  are  very  nearly  the  same,  and  if  the  two  orbits  are  of 
the  same  body,  the  equation 


298 


STABILITY   OF   PARTICULAR   SOLUTIONS. 


[160 


(27)      -±.  +  2^(1- 


cos 


=  —  +  2    a2(l  -  e22)  cos 


must  be  fulfilled,  where  the  elements  are  those  in  actual  use  by 
astronomers.  Such  is  the  criterion  developed  by  Tisserand,  and 
employed  later  by  Schulhof  and  others. 

160.  Stability  of  Particular  Solutions.  Five  particular  solutions 
of  the  motion  of  the  infinitesimal  body  have  been  found.  If  the 
infinitesimal  body  is  displaced  a  very  little  from  the  exact  points 
of  the  solutions  and  given  a  small  velocity  it  will  either  oscillate 
around  these  respective  points,  at  least  for  a  considerable  time, 
or  it  will  rapidly  depart  from  them.  In  the  first  case  the  particular 
solution  from  which  the  displacement  is  made  is  said  to  be  stable; 
in  the  second  case,  it  is  said  to  be  unstable. 

The  question  of  stability  must  be  formulated  mathematically. 
Consider  the  equations 


(28) 


Suppose  x  =  XQ,  y  =  y^  where  XQ  and  yQ  are  constants,  is  a  par- 
ticular solution  of  (28).     That  is, 

/(zo,  yo)  =  0,        g(x0,  yo)  =  0. 

Give  the  body  a  small  displacement  and  a  small  velocity  so  that 
its  coordinates  and  components  of  velocity  are 

x  =  xQ  +  x', 


(29) 


y  =  yo  + 

dx  =dx^ 
dt  ==  dt  ' 

dy  =  dyf 
dt  "  dt  > 


y', 


where  x',  y',  —  ,  and  -^-  are  initially  very  small.     On  making 
these  substitutions  in  (28),  the  differential  equations  become 


(30) 


160]  STABILITY    OF    PARTICULAR   SOLUTIONS.  299 

When  the  right  members  are  developed  by  Taylor's  formula,  they 
take  the  form 


2/0  -  g(xo,  2/o)  +       o/  + 


In  the  partial  derivatives  x  =  x0  and  y  =  y0.  The  first  terms  in 
the  right  members  are  respectively  zero;  hence  equations  (30) 
become 

r  ^~'       9  dy'  _  df    ,       df    , 

(31) 


If  #'  and  2/'  are  taken  very  small  on  the  start  the  influence  of 
the  higher  powers  in  the  right  members  will  be  inappreciable,  at 
least  for  a  considerable  time.  If  the  parts  which  involve  second 
and  higher  degree  terms  in  x'  and  yr  are  neglected,  the  differential 
equations  reduce  to  the  linear  system 


(32) 


_<2.=      _  x>    -          tf 
*      "          d 


___ 
dt   ~  dx'  dy'' 


The  solutions  of  a  system  of  linear  differential  equations  with  con 
stant  coefficients  can  in  general  be  expressed  in  terms  of  exponen 
tials  in  the  form 


where  ai,  •  •  •  ,  "<*4  are  the  constants  of  integration,  and  0i,  •  •  •  ,  04 
are  constants  depending  upon  them  and  the  constants  involved  in 
the  differential  equations.  If  Xi,  •  •  •  ,  X4  are  pure  imaginary 
numbers,  then  x'  and  yf  are  expressible  in  periodic  functions,  and 
the  solution  from  which  the  start  was  made  is  said  to  be  stable;  if 
any  of  Xi,  •  •  •  ,  X4  are  real  or  complex  numbers,  then  x'  and  y' 
change  indefinitely  with  t,  and  the  solution  is  said  to  be  unstable. 
There  are  exceptional  cases  where  the  solution  contains  constant 

terms  instead  of  exponentials;  they  are  of  course  stable  if  all  the 

— 


300 


APPLICATION    OF    CRITERION   FOR   STABILITY 


[161 


exponentials  are  purely  imaginary.  There  are  other  exceptional 
cases  in  which  the  solution  contains  exponentials  multiplied 
by  some  power  of  t\  these  solutions  are  usually  regarded'  as 
unstable. 

161.  Application  of  the  Criterion  for  Stability  to  the  Straight 
Line  Solutions.  The  definitions  and  general  methods  of  the  last 
article  will  now  be  applied  to  the  special  cases  which  have  arisen 
in  the  discussion  of  the  motion  of  the  infinitesimal  body.  The 
original  differential  equations  were  (Art.  152) 


d?x 


(x  — 


(x  — 


dx 
dt 


dt2 


-  (1  -  M)      ~  M      =  h(x,  y,  z). 


The  straight  line  solutions  occur  for 

x  =  x0i,        y  =  0,        z  =  0, 

where  i  =  1,  2,  3  according  as  the  point  lies  between  +  oo  and  /z, 
IJL  and  1  —  M,  or  1  —  /z  and  —  oo ,  and  where  these  values  of  x,  y, 
and  z  satisfy  equation  (19).  Make  the  substitution 


_ 
dt' 


X  =  Xoi  +  X1, 

y  =  y'} 

z 

dx      dx' 
dt==W' 

dy      dy' 
dt  "  dt  ' 

dz 
dt 

Then  it  is  found  that 


*.  4-       £   .  x 

*  ^27  ^2 


uy    i  i    uy    t  i  uy    /  __    /  _ 

dx'  dy'           dz' 

dh_    ,  dh_    ,       dh^  ,  = 

dx'X  dy'y        dz'Z 


(i  -  M 


Let 

(34) 


Then  the  equations  corresponding  to  (32)  become  in  this  case 


161] 


TO    THE    STRAIGHT  LINE    SOLUTIONS. 


301 


(35) 


df 


The  last  equation  is  independent  of  the  first  two  and  can  be 
treated  separately.     The  solution  is  (Art.  32) 


(36) 


z>  = 


Therefore  the  motion  parallel  to  the  z-axis,  for  small  displace 
ments,  is  periodic  with  the  period  — =  . 

4Ai 

Consider  now  the  simultaneous  equations 


(37) 


To  find  the  solutions  let 

(38) 

where  K  and  L  are  constants.     On  substituting  these  expressions 
in  equations  (37)  and  dividing  out  e^,  it  is  found  that 

[X2  -  (1  +  2Ai)]K  -  2XL  =  0, 
2\K  +  [X2  -  (1  -  Ai)]L  =  0. 


(39) 


In  order  that  equations  (38)  shall  be  particular  solutions  of  (37) 
equations  (39)  must  be  fulfilled.  They  are  verified  by  K  =  0, 
L  =  0;  but  in  this  case  x'  =  0,  yf  =  0,  and  the  solutions  reduce 
to  the  straight  line  solutions.  Equations  (39)  can  be  satisfied  by 
values  of  K  and  L  different  from  zero  only  if  the  determinant 
of  the  coefficients  vanishes.  This  condition  is 


(40) 


X2  -  (1  +  2A,), 
+  2X 


-  2X 

X2  -  (1  -  A,) 


0. 


This  equation  is  the  condition  upon  X  that  equations  (38)  may  be 
a  solution  of  (37).     There  are  four  roots  of  this  biquadratic,  each 


302  PARTICULAR   VALUES   OF   THE    CONSTANTS.  [162 

giving  a  particular  solution,  and  the  general  solution  is  the  sum 
of  the  four  particular  solutions  multiplied  by  arbitrary  constants; 
that  is,  if  the  four  roots  of  (40)  are  Xi,  X2,  X3,  X4,  the  general  solu- 
tion is 

I    Ju  J\~\\s  |     J-\- 2^  T~   -**- 3^  *"       1      ./V  4&       9 


where  the  K3-  are  the  arbitrary  constants  of  integration,  and  the 
LJ  are  denned  in  terms  of  them  respectively  by  either  of  the 
equations  (39).  The  X,  depend  of  course  upon  the  subscript  i  on 
A,  but  the  notation  need  not  be  burdened  with  this  fact  since  the 
equations  all  have  the  same  form  whether  i  is  1,  2,  or  3. 

It  remains  to  determine  the  character  of  the  roots  of  the  bi- 
quadratic (40).  It  follows  from  (34)  and  (21),  (22),  and  (24) 
respectively  that 


(42) 


-     1  "A*      -    M  _  4_o  .  S^V 
~(l+r2)^  +  r23~  Z     3\3y 


4-o^+^  =  4  +  2-3(iV 

1  -M  M 

A3-(l-^+ (2^7)5-    l,*««* 


It  follows  from  (42)  that,  for  small  values  of  M>  the  term  of  (40) 
which  is  independent  of  X  satisfies  the  inequality 

1  +  A.--2A;2  <0,        (i  =  1,  2,  3); 

and,  indeed,  this  relation  is  true  for  values  of  M  up  to  the  limit  J, 
as  can  be  verified  easily.*  Therefore  the  biquadratic  has  two  real 
roots  which  are  equal  in  numerical  value  and  opposite  in  sign,  and 
two  conjugate  pure  imaginaries.  It  follows  from  the  definitions 
given  that  the  motion  is  unstable.  If  the  infinitesimal  body  were 
displaced  a  very  little  from  the  points  of  solution  it  would  in 
general  depart  to  a  comparatively  great  distance. 

162.  Particular  Values  of  the  Constants  of  Integration.  The 
constants  of  integration  will  now  be  expressed  in  terms  of  the 
initial  conditions,  and  it  will  be  shown  that  the  latter  can  be 
selected  so  that  the  motion  will  be  periodic. 

Suppose  Xi  and  X2  are  the  real  roots  of  equation  (40);  then 
^i  =  —  X2.  The  imaginary  roots  are 

*  H.  C.  Plummer  gave  a  general  proof  in  Monthly  Not.  of  Roy.  Astr.  Soc., 
vol.  LXII.  (1901). 


162] 


PARTICULAR   VALUES    OF   THE    CONSTANTS. 


303 


where  a  is  a  real  number.     The  Lj  are  expressed  in  terms  of  the 
KJ  by  equations  (39),  and  are 


(43) 


[V  -  (1  +  2AJ] 

"  -- 


'  =  1,2,  3; 
«  1,2,  3, 


Since  the  X,-  are  equal  in  numerical  value  but  opposite  in  sign  in 
pairs,  and  the  last  two  are  imaginary,  it  follows  that 


Ci  =    —  C2, 


(44) 

Af      1  c, 
where  c  is  a  real  constant  depending  on  i. 

Let  XQ,  2/0',  ~^~  ,  and  -~  be  the  initial  coordinates  and  com- 
ponents of  velocity;  then  equations  (41)  give  at  t  =  0 


-K2)+ 


dxj 
dt 


dt 


The  values  of  the  constants  of  integration  are  found  in  terms  of 
the  initial  coordinates  and  components  of  velocity  by  solving  these 
equations. 

The  values  of  x'  and  yr  increase  in  general  without  limit  with  the 
time,  but  if  the  initial  conditions  are  such  that  KI  =  K<>  =  0  they 
become  purely  periodic.  This  case  will  now  be  considered.  The 
initial  coordinates,  XQ,  y0',  will  determine  Ks  and  K^  by  means 

of  which  ~~  and  ~-  are  defined.     Thus 
at  dt 


whence 


304 


PARTICULAR  VALUES   OF   THE    CONSTANTS. 


[162 


2c 


The  equations  (41)  become 


(45) 


2c 


yo  • 

=  XQ  cos  at  +  —  sin  at, 
c 


=  —  coV  sin  at  +  2/0'  cos  <r£. 

The  equation  of  the  orbit  is  found  by  eliminating  t  from  these 
equations.  Solve  for  cos  at  and  sin  at;  then  square  and  add,  and 
the  result,  after  dividing  out  common  factors,  is 


(46) 


1. 


c2 


This  is  the  equation  of  an  ellipse  with  the  major  and  minor  axes 
lying  along  the  coordinate  axes,  and  with  the  center  at  the  origin. 
Since  X3  is  imaginary  it  follows  from  (43)  and  (44)  that  c2  >  1 ; 
therefore  the  major  axis  of  the  ellipse  is  parallel  to  the  7/-axis. 
The  eccentricity  is  given  by 


which,  for  large  values  of  c,  is.  very  near  unity.  The  orbits  have 
the  remarkable  property  that  their  eccentricity  is  independent 
of  the  initial  small  displacements,  depending  only  upon  the  dis- 
tribution of  the  mass  between  the  finite  bodies,  and  upon  the  one 
of  the  three  straight  line  solutions  from  which  they  spring. 

It  is  obvious  that  this  discussion  is  not  completely  rigorous 
because  the  terms  of  higher  degree  in  the  right  members  of  the 
differential  equations  have  been  neglected.  The  linear  terms 
alone  do  not  give  sufficient  conditions  for  the  existence  of  periodic 
orbits,  and  consequently  when  the  discussion  is  thus  restricted  it 
answers  only  the  question  as  to  the  stability  of  the  solution.  But 
in  the  present  case  periodic  orbits  actually  exist  about  all  three 


163]  APPLICATION   TO   THE    GEGENSCHEIN.  305 

points  for  all  0  <  M  ^  ^.  Some  special  examples  for  JJL  =  ^  were 
found  by  Darwin  in  his  memoir  in  Ada  Mathematica,  vol.  21. 
The  complete  analysis  for  these  orbits,  including  the  much  more 
difficult  case  in  which  the  finite  bodies  describe  elliptical  orbits, 
was  given  by  the  author  in  the  Mathematische  Annalen,  vol. 
LXXIII.  (1912),  pp.  441-479,  and  in  the  Publications  of  the  Carnegie 
Institution  of  Washington,  No.  161,  Periodic  Orbits,  chapters  v., 
vi.,  and  vii. 

163.  Application  to  the  Gegenschein.  If  the  constants  KI 
and  KI  are  zero  the  infinitesimal  body  will  revolve  in  an  ellipse 
around  the  point  of  equilibrium.  If  these  constants  are  not  zero 
but  small  in  numerical  value  compared  to  K3  and  K4,  the  motion 
will  be  nearly  in  an  ellipse  for  a  considerable  time,  but  will  eventu- 
ally depart  very  far  from  it.  It  would  be  possible  to  have  any 
number  of  infinitesimal  bodies  revolving  around  the  same  point 
without  disturbing  one  another. 

Consider  the  motion  of  the  earth  around  the  sun.  It  is  in  a 
curve  which  is  nearly  a  circle.  One  of  the  straight  line  solution 
points  is  exactly  opposite  to  the  sun,  and  if  a  meteor  should  pass 
near  it  with  initial  conditions  approximately  such  as  have  been 
defined  in  the  last  article  it  would  make  one  or  more  circuits  around 
this  point  before  pursuing  its  path  into  other  regions.  If  a  very 
great  number  were  swarming  around  this  point  at  one  time  they 
would  appear  from  the  earth  as  a  hazy  patch  of  light  with  its  center 
at  the  anti-sun,  and  elongated  along  the  ecliptic.  This  is  the 
appearance  of  the  gegenschein  which  was  discovered  independently 
by  Brorsen,  Backhouse,  and  Barnard  in  1855,  1868,  and  1875 
respectively. 

The  crucial  question  seems  to  be  whether  or  not  there  are  enough 
meteors  with  the  approximate  initial  conditions  to  explain  the 
observed  phenomena,  but  no  certain  answer  can  be  given.  How- 
ever, it  is  certain  that  the  meteors  are  exceedingly  numerous,  as 
many  as  8,000,000  striking  into  the  earth's  atmosphere  daily 
according  to  H.  A.  Newton;  and  it  is  only  reasonable  to  sup- 
pose that  they  cause  the  zodiacal  light  which  is  very  bright  com- 
pared to  the  gegenschein.  The  suggestion  that  this  may  be  the 
cause  of  the  gegenschein  was  first  made  by  Gylden  in  the  closing 
paragraph  of  a  memoir  in  the  Bulletin  Astronomique,  vol.  i.,  en- 
titled, Sur  un  Cas  Particulier  du  Probleme  des  Trois  Corps.* 

*  See  also  a  paper  by  F.  R.  Moulton  in  The  Astronomical  Journal,  No.  483. 
21 


306 


APPLICATION    OF    CRITERION    FOR   STABILITY 


[164 


164.  Application  of  the  Criterion  for  Stability  to  the  Equilateral 
Triangle  Solutions.  The  particular  solutions  of  the  original  differ- 
ential equations  in  this  case  are  rx  =  1,  r2  =  1.  The  equations 
corresponding  to  (33)  are 


bA 


dh    f       dh    ,       dh    f  _          , 

x  *v         z       ~z' 


and  the  differential  equations  up  to  terms  of  the  second  degree  are 


(47) 


The  last  equation  is  independent  of  the  first  two,  and  its  solution  is 
z'  =  d  sin  t  +  Cz  cos  t. 

Therefore  the  motion  parallel  to  the  2-axis,  for  small  displace- 
ments, is  periodic  with  period  2ir,  the  same  as  that  of  the  revo- 
lution of  the  finite  bodies. 
To  find  the  solutions  of  the  first  two  equations  let 


(48) 


fx'  =  Ke", 
\y'  =  Le». 


On  substituting  these  expressions  in  the  first  two  equations  of  (47) 
and  dividing  out  common  factors,  it  is  found  that 


(49) 


[X2- 


-  2M)     L  =  0, 


2\  - 


K 


f  ]L  =  0. 


In  order  that  solutions  may  be  obtained  other  than  xf  =  0,  y'  =  0 
the  determinant  of  these  equations  must  vanish.     That  is, 


164] 


TO   EQUILATERAL  TRIANGLE   SOLUTIONS. 


307 


(50) 


X2  -  i,  -  2X  - 


(1  -  2/*),  X2  -  f 


Let  Xi,  X2,  X3,  X4  be  the  roots  of  this  biquadratic.     Then  the 
general  solutions  of  (47)  are 


x   = 
y>  = 


where  KI,  1^2,  -^3,  ^4  are  the  constants  of  integration,  and  LI,  L2, 
L3,  L4  are  constants  related  to  them  by  either  of  equations  (49)  . 
It  is  found  from  (50)  that 


Xi  =  -  X2  = 


-  1+A/l  - 


-M). 


-  1  -  Vl  -  27^(1  - 


The  number  /i  never  exceeds  £>  an<i  if  1  —  27/x(l  —  /z)  ^  0  the 
roots  are  pure  imaginaries  in  conjugate  pairs;  if  this  inequality 
is  not  fulfilled  they  are  complex  quantities.  The  inequality  may 
be  written 

1  -  27M(1  -  M)  =  e, 

where  e  is  a  positive  quantity  whose  limit  is  zero.  The  solution  of 
this  equation  is 


Since  ju  represents  the  mass  which  is  less  than  one-half  the  negative 
sign  must  be  taken.  At  the  limit  €  =  0,  /i  =  .0385  •  •  •  .  There- 
fore if  ju  <  .0385  •  •  •  the  roots  of  (50)  are  pure  imaginaries  and 
the  equilateral  triangle  solutions  are  stable  ;  if  ju  >  .0385  •  •  •  the 
roots  of  (50)  are  complex  and  the  equilateral  triangle  solutions 
are  unstable. 

XXI.     PROBLEMS. 

1.  If  a  comet  approaching  the  sun  in  a  parabola  should  be  disturbed  by 
Jupiter  so  that  its  orbit  remained  a  parabola  while  its  perihelion  distance  was 
doubled,  what  would  be  the  relation  between  the  new  inclination  and  the  old? 


Ans. 


COS 


V2 

=  -—  COS 

2 


308  PROBLEMS. 

2.  Prove  that  if  a  comet's  orbit,  whose  inclination  to  Jupiter's  orbit  is 
zero,  is  changed  by  the  perturbations  of  Jupiter  from  a  parabola  to  an  ellipse 
the  parameter  of  the  orbit  is  necessarily  decreased.     Investigate  the  changes 
in  the  parameters  for  changes  in  the  major  axes  of  the  other  species  of  conies. 

3.  Suppose  a  comet  is  moving  in  an  ellipse  in  the  plane  of  Jupiter's  orbit 
and  that  the  perturbing  action  of  Jupiter  is  inappreciable  except  for  a  short 
time  when  they  are  near  each  other.     Prove  that  if  the  perturbation  of  Jupiter 
has  increased  the  eccentricity,  the  period  has  been  increased  or  decreased 
according  as  the  product  of  the  major  semi-axis  and  the  square  root  of  the 
parameter  in  the  original  ellipse  is  greater  or  less  than  unity  when  expressed 
in  the  canonical  units. 

4.  A  particle  placed  midway  between  two  equal  fixed  masses  is  in  equilib- 
rium.    Investigate  the  character  of  the  equilibrium  by  the  method  of  Art.  161. 

5.  Suppose  1  —  fj,  and  fj.  are  the  sun  and  earth  respectively;  find  the  period 
of  oscillation  parallel  to  the  z-axis  for  an  infinitesimal  body  slightly  displaced 
from  the  xy-plaue  near  the  straight  line  solution  point  opposite  to  the  sun 
with  respect  to  the  earth  as  an  origin. 

Ans.     183.304  mean  solar  days. 

6.  In  the  same  case,  find  the  period  of  oscillation  in  the  xy-pl&ne. 
Ans.     ISfaft  mean  solar  days. 

1~7  k 

7.  Prove  that  in  general  for  small  values  of  /JL  the  periods  of  oscillation 
both  parallel  to  the  z-axis  and  in  the  xy-plane,  are  longest  for  the  point  opposite 
to  n  with  respect  to  1  —  n  as  origin;  next  longest  for  the  point  opposite  to 
1  —  n  with  respect  to  n  as  origin;  and  shortest  for  the  point  between  1  —  n 
and  fj,. 

8.  Find  the  eccentricity  of  the  orbit  in  the  xy-plane  opposite  to  the  sun  in 
the  case  of  the  sun  and  earth. 

9.  The  differential  equations  (35)  admit  the  integral 


discuss  the  meaning  of  this  integral  after  the  manner  of  articles  154-159. 

10.  What  can  be  said  regarding  the  independence  of  equations  (39)  after 
the  condition  has  been  imposed  that  the  determinant  shall  vanish? 

11.  If  the  explanation  of  the  gegenschein  given  in  Art.  163  is  true  what 
should  be  its  maximum  parallax  in  celestial  latitude  for  an  observer  in  lati- 
tude 45°? 

Ans.     Roughly  15'.     (Too  small  to  be  observed  with  certainty  in  such  an 
indefinite  object.) 

12.  Suppose  /z  =  \  and  reduce  the  problem  of  finding  the  motion  of  the 
infinitesimal  body  through  the  origin  along  the  z-axis  to  elliptic  integrals. 


165] 


CONDITIONS   FOR   CIRCULAR  ORBITS. 


309 


CASE  OF  THREE  FINITE  BODIES. 

165.  Conditions  for  Circular  Orbits.  The  theorem  of  Lagrange 
that  it  is  possible  to  start  three  finite  bodies  in  such  a  manner 
that  their  orbits  will  be  similar  ellipses,  all  described  in  the  same 
time,  will  be  proved  in  this  section.  It  will  be  established  first 
for  the  special  case  in  which  the  orbits  are  circles.  It  will  be 
assumed  that  the  three  bodies  are  projected  in  the  same  plane. 
Take  the  origin  at  their  center  of  mass  and  the  ^-plane  as  the 
plane  of  motion.  Then  the  differential  equations  of  motion  are 
(Art.  143) 

—  =  —  (i  =  1     2    ^] 


(52) 


ldU 


dt2   ~ 
U  = 


The  motion  of  the  system  is  referred  to  axes  rotating  with  the 
uniform  angular  velocity  n  by  the  substitution 


(53) 


£i  =  Xi  cos  nt  —  yi  sin  nt, 
rji  =  Xi  sin  nt  +  yi  cos  nt. 


(i  =  1,  2,  3), 


On  making  the  substitution,  and  reducing  as  in  Art.  152,  it  is 
found  that 


(54) 


d2yi  .    0    dxf  1   dU      n 

-~  +  2?i  -r  —  n2t  --  -—  =0 


dt2 


j 
dt 


If  the  bodies  are  moving  in  circles  around  the  origin  with  the 
angular  velocity  n}  their  coordinates  with  respect  to  the  rotating 
axes  are  constants.  Since  the  first  and  second  derivatives  are 
then  zero,  equations  (54)  become 


(55) 


I,  2 


L,  2 


I,  3 


2,  3 


2,  3 


310 


EQUILATERAL  TRIANGLE   SOLUTIONS. 


[166 


(55) 


+ 


+ 


^2, 


=  0, 
=  0, 
=  0. 


And  conversely,  if  the  masses  and  initial  projections  are  such 
that  these  six  equations  are  fulfilled  the  bodies  move  in  circles 
around  the  origin  with  the  uniform  angular  velocity  n. 

Since  the  origin  is  at  the  center  of  mass  the  coordinates  satisfy 


(56) 


+ 


+ 


=  0, 
=  0. 


If  the  first  equation  of  (55)  is  multiplied  by  mi,  the  second  by  ra2, 
and  the  products  added,  the  sum  becomes,  as  a  consequence  of 
the  first  equation  of  (56),  the  third  of  (55).  In  a  similar  manner 
the  last  equation  of  (55)  can  be  derived  from  the  others  in  y  and  the 
last  of  (56).  Therefore  the  third  and  sixth  equations  of  (55)  can 
be  suppressed,  and  equations  (56)  used  in  place  of  them,  giving  a 
somewhat  simpler  system  of  equations. 

The  units  of  time,  space,  and  mass  are  so  far  arbitrary.  It  is 
possible,  without  loss  of  generality,  to  select  them  so  that  ri,  2  =  1 
and  k2  =  1.  Then  necessary  and  sufficient  conditions  for  the 
existence  of  solutions  in  which  the  orbits  are  circles  are 


(57) 


=  0. 


166.  Equilateral  Triangle  Solutions.  There  is  a  solution  of  the 
problem  for  every  set  of  real  values  of  the  variables  satisfying 
equations  (57).  It  is  easy  to  show  that  the  equations  are  fulfilled 


167] 


STRAIGHT   LINE    SOLUTIONS. 


311 


if  the  bodies  lie  at  the  vertices  of  an  equilateral  triangle. 
PI,  2  =  TZ,  3  =  ri,  3  =  1,  and  equations  (57)  become 

-f-  m3#3  =  0, 


Then 


(m2 
(mi 

(m2 


m3  — 


m3  - 


m3  —  n2)y 


=  0, 
=  0, 

+  m2?/2  +  m3?/3  =  0, 
-  m2?/2  -  msy3  =  0, 
0. 


These  equations  are  linear  and  homogeneous  in  x\,  Xz,  -  •  •  ,  2/3. 
In  order  that  they  may  have  a  solution  different  from  Xi  =  xz 
=  -  •  •  =2/3  =  0,  which  is  incompatible  with  7*1,  2  =  r2,  3  =  ri,  3  =  1, 
the  determinant  of  their  coefficients  must  vanish.  On  letting 
M  =  mi  -\-  mz  -\-  m3,  it  is  easily  found  that  this  condition  is 

m32(M  -  w2)4  =  0, 

from  which  ri2  =  M.  Then  two  of  the  z»  and  two  of  the  y<  are 
arbitrary,  and  hence  the  equations  have  a  solution  compatible 
with  rt,  /  =  1.  Therefore,  the  equilateral  triangular  configuration 
with  proper  initial  components  of  velocity  is  a  particular  solution  of 
the  Problem  of  Three  Bodies;  and,  if  the  units  are  such  that  the 
mutual  distances  and  k2  are  unity,  the  square  of  the  angular  velocity 
of  revolution  is  equal  to  the  sum  of  the  masses  of  the  three  bodies. 

167.  Straight  Line  Solutions.  The  last  three  equations  of  (57) 
are  fulfilled  by  y\  —  yz  =  y3  =  0,  that  is,  if  the  bodies  are  all  on  the 
x-axis.  Suppose  they  lie  in  the  order  mi,  m2,  m3  from  the  negative 
end  of  the  axis  toward  the  positive.  Then  x3  >  x2  >  x\  and 
TI,  2  =  Xz  —  x\  =  1,  and  the  first  three  equations  of  (57)  become 


(58) 


=  0, 


m3 


1   («,  -  zi  -  I)2 
On  eliminating  z3  and  n2,  it  is  found  that 

(59)     m2  -|-  (mi  +  mz)xi  +  7 


=  0. 


1    (Mzi  +  m2)2 
If  this  equation  is  cleared  of  fractions  a  quintic  equation  in  x\  is 


312  DYNAMICAL   PROPEETIES   OF   SOLUTIONS.  [168 

obtained  whose  coefficients  are  all  positive.  Therefore  there  is 
no  real  positive  root  but  there  is  at  least  one  real  negative  root, 
and  consequently  at  least  one  solution  of  the  problem. 

Instead  of  adopting  xi  as  the  unknown,  #3  —  z2,  which  will  be 
denoted  by  A,  may  be  used.  The  distance  Xi  must  be  expressed 
in  terms  of  this  new  variable.  The  relations  among  x\,  x2,  xz, 
and  A  are 

I  rv 

Xz  —  Xi  =  1, 

_    _         A     . 

3/3  "^  *C2   ~~  ^i  , 

whence 


M 

On  substituting  this  expression  for  Xi  in  (59),  clearing  of  fractions, 
and  dividing  out  common  factors,  the  condition  for  the  collinear 
solutions  becomes 

(mi  +  m2)A5  +  (3mi  +  2m2)A4  +  (3mi  - 


(60) 

-  (ra2  +  3m3)A2  -  (2m2  +  3w3)A  -  (m2  +  ma)  =  0. 

This  is  precisely  Lagrange's  quintic  equation  in  A,*  and  has  but 
one  real  positive  root  since  the  coefficients  change  sign  but  once. 
The  only  A  valid  in  the  problem  for  the  chosen  order  of  the  masses 
is  positive;  hence  the  solution  of  (60)  is  unique  and  defines  the 
distribution  of  the  bodies  in  the  straight  line  solution  of  the 
Problem  of  Three  Bodies.  It  is  evident  that  two  more  distinct 
straight  line  solutions  will  be  obtained  by  cyclically  permuting 
the  order  of  the  three  bodies. 

168.  Dynamical  Properties  of  the  Solutions.  Since  the  bodies 
revolve  in  circles  with  uniform  angular  velocity  around  the  center 
of  mass,  the  law  of  areas  holds  for  each  body  separately;  therefore 
the  resultant  of  all  the  forces  acting  upon  each  body  is  constantly 
directed  toward  the  center  of  mass  (Art.  48). 

Let  the  distances  of  mi,  w2,  and  ra3  from  their  center  of  mass 
be  ai,  a2,  and  a3  respectively.  Then  the  centrifugal  acceleration 

V 2 
to  which  mt-  is  subject  i&«»  =  — -  ,  where  Vt-  is  the  linear  velocity 

Q/i 

of  m».     But  this  may  be  written  on  =  tfai.     The  centripetal  force 

*  See  Lagrange's  Collected  Works,  vol.  vi.,  p.  277,  and  Tisserand's  Mec.  Cel., 
vol.  i.)  p.  155. 


169] 


GENERAL    CONIC    SECTION    SOLUTION. 


313 


exactly  balances  the  centrifugal;  therefore  the  acceleration  toward 
the  center  of  mass  is 

cti  =  n2di', 

that  is,  the  accelerations  of  the  various  bodies  toward  their  common 
center  of  mass  are  directly  proportional  to  their  respective  distances 
from  this  point. 

169.  General  Conic  Section  Solutions.  The  solutions  of  the 
problem  of  three  bodies  which  have  been  discussed  are  char- 
acterized by  the  fact  that  their  orbits  are  circles.  It  will  be  shown 
that  corresponding  to  each  of  them  there  is  a  solution  in  which 
the  orbits  are  conic  sections  of  arbitrary  eccentricity.  These 
solutions  are  characterized  by  the  fact  that  in  them  the  ratios  of 
the  mutual  distances  of  the  bodies  are  constant,  though  the  dis- 
tances themselves  are  variable. 

The  differential  equations  of  motion  when  the  system  is  referred 
to  fixed  axes  with  the  origin  at  the  center  of  gravity  of  the  system 
are 


(61) 


dP 


• 


dt2 
~W 
W 
~W 

w 


-m) 


-u») 


-10 


2,3 


Wi(r?3  —  171) 


Suppose  the  coordinates  of  m\,  m2,  and  w3  at  i  =  £0  are  respec- 
tively (a?i,  2/1),  (a?2,  2/2),  and  (x3,  2/3),  and  let  the  respective  distances 
from  the  origin  be  ri(%  r2(0),  and  r3(0).  Suppose  the  angles  that 
ri(0),  r2(0),  and  r3(0)  make  with  the  ^-axis  are  <pi,  <p2,  and  <p3.  Then 


(62) 


<„- 

[yi  = 


=  ri(0)  cos 


r2(0)  cos  (pz, 


ri(0)  sin 


=  r2(0)  sin 


=  r*3(0)  cos 
>(0>  sin 


2/3  =  r3^ 

Now  let  the  coordinates  of  the  bodies  at  any  time  t  be  (£1,  771), 
(£2,  *?2),  and  (£3,  rjs).     Suppose  the  ratios  of  the  mutual  distances 


314  GENERAL    CONIC   SECTION   SOLUTIONS. 

are  constants;  then  the  mutual  distances  at  t  are 


[169 


,  3, 


where  p  is  the  factor  of  proportionality.     Since  the  shape  of  the 
figure  formed  by  the  three  bodies  is  unaltered,  it  follows  that 

(63)  ri  =  ri(0)p, 


=  r2(0)p, 


Fig.  42. 

Moreover,  the  radii  n,  r2,  and  r3  will  have  turned  through  the  same 
angle  6.     Hence 


(64)  < 


£1  =  n(0)P  cos  (B  + 

??i  =  ri(0)p  sin  (0  + 

£2  =  r2(0)p  cos  (0  + 

??2  =  r2(0)p  sin  (0  + 

^3  =  r3(0)p  cos  (0  + 

773  =  r3(0)p  sin  (0  + 


=  (zi  cos  6  -  2/1  sin  0)p, 

=  (0:1  sin  0  +  2/1  cos  0)p, 

=  (x2  cos  6  —  2/2  sin  0)p, 

=  (a;2  sin  0  +  2/2  cos  0)p, 

=  (x3  cos  0  —  2/3  sin  0)p, 

=  (z3  sin  0  +  2/3  cos  0)p. 


If  equations  (61)  are  transformed  by  means  of  (64)  they  will 
involve  only  the  two  dependent  variables  p  and  0,  and  they  will 
be  necessary  conditions  for  the  existence  of  solutions  in  which  the 
ratios  of  the  mutual  distances  are  constants.  It  follows  from 
the  first  two  equations  of  (61)  and  (64)  after  multiplying  the  results 


169] 


GENERAL    CONIC   SECTION   SOLUTIONS. 


315 


of  the  transformation  by  cos  6  and  sin  0  and  adding,  and  then  by 
—  sin  0  and  cos  0  and  adding,  that 

dd\2  d*e 


(65)  < 


Let 
(66) 
Then 


f 


. 


de 


de\2 


dp  dS 
dt  dt 


I  mz(yi  -  yd      m3(j/i  -  2/3)  1  1_ 
"  I  "     r\  2  r\  3        J  P2 ' 


,dO 


P  dt' 


and  equations  (65)  become 


(67) 


dt2 


_ 
dt       p3 


x\ 


r3i, 


2/ip 


1 


~  3/2)  , 
,  2 


r3i,3 
IT1"  JP1' 


And  the  equations  which  are  similarly  derived  from  the  last  four 
equations  of  (61)  and  of  (65)  are 


(68)^ 


__ 
dt2       x2p  dt       p3 


r31> 


,  2 


dt2 


eft       p3 


k{ 
i{ 


. 


+ 


r3 


2,3 


\  1 
Jp2' 


1,3 


, 


-2/2)U 

~   I  ~2 

J,  3  J  P 


Equations  (67)  and  (68)  are  necessary  conditions  for  the  exist- 
ence of  solutions  in  which  the  ratios  of  the  distances  of  the  bodies 
are  constants.  There  are  but  two  variables,  p  and  ^,  to  be  de- 
termined. The  first  gives  the  dimensions  of  the  system  by  means 
of  (63) ,  and  the  second  its  orientation  by  means  of  (66) .  In  order 


316 


GENERAL    CONIC   SECTION   SOLUTIONS. 


[169 


that  the  solutions  in  question  may  exist  these  equations  must  be 
consistent.  In  pairs  of  two  they  define  p  and  ^  when  the  initial 
conditions  are  specified.  In  order  that  for  given  initial  con- 
ditions the  p  and  \J/  shall  be  identical  as  defined  by  each  of  the 
three  pairs  of  differential  equations,  the  coefficients  of  corre- 
sponding terms  in  p  and  ^  must  be  the  same.  This  can  be  proved 
by  considering  the  expansion  of  the  solutions  as  power  series  in 
t  —  to  by  the  method  of  Art.  127.  In  order  that  the  solutions 
shall  be  the  same  the  coefficients  of  corresponding  powers  of 
t  —  t0  must  be  identical;  and  in  order  that  these  conditions  shall 
be  satisfied  the  coefficients  of  corresponding  terms  in  the  differ- 
ential equations  must  be  identical.  Therefore  the  conditions  for 
the  consistency  of  equations  (67)  and  (68)  are  either 


(69) 


or 


(70) 


dt 


2/3 


=  0, 


and  the  system  of  six  equations 


(71) 


+ 


+ 


•* 


,., 


+ 


™2(2/i  -  2/2) 


+ 


=  n2 


if, 


+ 


+ 


o 

^ 


2,  3 

usi-.rf 


where  n2  is  the  common  constant  value  of  the  brackets  in  the  right 
members  of  (67)  and  (68).  And  it  follows  from  equations  (71), 
as  well  as  from  the  original  definitions  of  the  Xi  and  the  yiy  that 
the  center  of  mass  equations 

f  m'lXi  -f-  m2z2  -f  ^30:3  =  0, 

I  mlyl  -f-  mzy2  +  m3y3  =  0, 
are  fulfilled. 


169]  GENERAL   CONIC   SECTION   SOLUTIONS.  317 

Equations  (69)  are  satisfied  only  if  the  three  bodies  are  in  a 
straight  line  at  i  —  tQ.  Since,  by  hypothesis,  the  shape  of  the 
configuration  is  constant,  they  always  remain  in  a  straight  line 
in  this  case.  The  position  of  the  axes  can  be  so  chosen  at  t  =  t0 
that  2/1  =  2/2  =  2/3  =  0  and  the  conditions  for  the  existence  of  the 
solutions  reduce  to  the  first  three  equations  of  (71).  These 
equations  are  the  same  as  (55)  of  Art.  165,  and  it  was  shown 
in  Art.  167  that  they  have  but  three  real  solutions. 

Suppose  equations  (69)  are  satisfied  and  that  the  bodies  remain 
collinear;  therefore  the  resultant  of  all  the  forces  to  which  each 
one  is  subject  is  directed  constantly  toward  the  center  of  gravity 
of  the  system,  and  consequently  the  law  of  areas  with  respect  to 
this  point  holds.  Hence 


where  Ci,  0%,  and  Ca  are  constants.     It  follows  from   (63)   that 
p2  —  =  .    (*     ,  and  then  from  (66)  that 
(66),  (67),  and  (68)  become  in  this  case 


p2  —  =  .    (*     ,  and  then  from  (66)  that  ~  =  0.      Hence  equations 


(72) 


\l/  =  Co  =  constant, 


_ 
di 


These  are  the  differential  equations  in  polar  coordinates  for  the 
Problem  of  Two  Bodies.  Except  for  differences  of  notation,  they 
are  the  same  as  equations  (65)  of  chap.  v.  Therefore  p  and  8 
satisfy  the  conditions  of  conic  section  motion  under  the  law  of 
gravitation,  and  it  follows  from  (63)  and  the  definition  of  6  that  the 
three  bodies  describe  similar  conic  sections  having  an  arbitrary 
eccentricity.  These  solutions  include  the  straight  line  solutions 
in  which  the  orbits  are  circles  as  a  special  case. 

Suppose  equations  (69)  are  not  satisfied;  then  the  bodies  are 
not  collinear.  But  if  the  bodies  are  not  collinear  equation  (70) 
must  hold  in  order  that  equations  (67)  and  (68)  may  be  com- 
patible. It  follows  from  equations  (66)  and  (63)  that  the  law  of 
areas  with  respect  to  the  origin  holds  for  each  body  separately. 
It  was  shown  in  Art.  166  that  equations  (71)  are  satisfied  if  the 


318  PROBLEMS. 

bodies  are  at  the  vertices  of  an  equilateral  triangle.  It  is  easy  to 
show  that,  unless  they  are  collinear,  there  is  no  other  solution. 
In  the  case  of  the  equilateral  triangle  solution  equations  (67)  and 
(68)  also  reduce  to  (72),  and  the  orbits  are  similar  conic  sections 
of  arbitrary  eccentricity. 

XXII.     PROBLEMS. 

1.  Take  as  an  hypothesis  that  a  solution  exists  in  which  the  three  bodies 
are  always  collinear.     Prove  that  the  law  of  areas  holds  for  each  body  sepa- 
rately with  respect  to  the  center  of  mass  of  the  system,  with  respect  to  either 
of  the  other  bodies,  and  with  respect  to  the  center  of  mass  of  any  two  of  the 
bodies. 

2.  Write  the  conditions  that  the  accelerations  to  which  the  bodies  are 
subject  shall  be  directed  toward  their  common  center  of  mass  and  proportional 
to  their  respective  distances. 

Ans.    Equations  (55). 

3.  The  resultant  of  the  forces  acting  on  each  body  always  passes  through 
a  fixed  point.     Prove  that  the  equilateral  triangle  configuration  is  the  only 
solution  of  equations  (55)  unless  the  bodies  lie  in  a  straight  line. 

4.  Suppose  nil  =  m2  =  m3  =  1,  and  that  the  bodies  move  according  to 
the  equilateral  triangular  solution.     Find  the  radius  of  the  circle  in  which  a 
particle  would  revolve  around  one  of  them  in  the  period  in  which  they  revolve 
around  their  center  of  mass. 

Ans.     R  =  3  *. 

5.  Prove  that  the  equilateral  triangular  circular  solutions  hold  when  the 
mutual  attractions  of  the  bodies  vary  as  any  power  of  the  distance. 

6.  Find  the  number  of  collinear  solutions  when  the  force  varies  as  any 
power  of  the  distance. 

7.  Prove  that  when  the  force  varies  inversely  as  the  fifth  power  one  solution 
is  that  each  of  the  bodies  moves  in  a  circle  through  their  center  of  mass  in 
such  a  way  that  the  three  bodies  are  always  at  the  vertices  of  an  equilateral 
triangle. 

8.  Prove  that  if  the  three  bodies  are  placed  at  rest  in  any  one  of  the  con- 
figurations admitting  circular  solutions,  they  will  fall  to  their  center  of  mass 
in  the  same  time  in  straight  lines. 

9.  Find  the  distribution  of  mass  among  the  three  bodies  for  which  the  time 
of  falling  to  their  center  of  mass  will  be  the  least;  the  greatest. 

10.  Prove  that  if  any  four  masses  are  placed  at  the  vertices  of  a  regular 
tetrahedron,  the  resultant  of  all  the  forces  acting  on  each  body  passes  through 
the  center  of  mass  of  the  four,  and  that  the  magnitudes  of  the  accelerations  are 
proportional  to  the  respective  distances  of  the  bodies  from  their  center  of  mass. 

11.  Prove  that  there  are  no  circular  solutions  in  the  Problem  of  Four 
Bodies  in  which  the  bodies  do  not  all  move  in  the  same  plane. 

12.  Investigate  the  stability  of  the  triangle  and  straight  line  solutions 
of  the  Problem  of  Three  Bodies  when  all  of  the  masses  are  finite. 


HISTORICAL   SKETCH.  319 


HISTORICAL  SKETCH  AND   BIBLIOGRAPHY. 

The  first  particular  solutions  of  the  Problem  of  Three  Bodies  were  found 
by  Lagrange  in  his  prize  memoir,  Essai  sur  le  Probleme  des  Trois  Corps,  which 
was  submitted  to  the  Paris  Academy  in  1772  (Coll.  Works,  vol.  vi.,  p.  229, 
Tisserand's  Mec.  Cel.  vol.  i.,  chap.  vin.).  The  solutions  which  he  found  are 
precisely  those  given  in  the  last  part  of  this  chapter.  His  method  was  to 
divide  the  problem  into  two  parts;  (a)  the  determination  of  the  mutual  dis- 
tances of  the  bodies,  (6)  having  solved  (a),  the  determination  of  the  plane 
of  the  triangle  in  space  and  the  orientation  of  the  triangle  in  the  plane.  He 
proved  that  if  the  part  (a)  were  solved  the  part  (6)  could  also  be  solved. 
To  solve  (a)  it  was  necessary  to  derive  three  differential  equations  involving 
the  three  mutual  distances  alone  as  dependent  variables.  He  found  three 
equations,  one  of  which  was  of  the  third  order,  and  the  remaining  two  of  the 
second  order  each,  making  the  whole  problem  of  the  seventh  order.  The  reduc- 
tion of  the  general  problem  of  three  bodies  by  the  ten  integrals  leaves  it  of  the 
eighth  order;  hence  Lagrange's  analysis  reduced  the  problem  by  one  unit.  He 
found  that  he  could  integrate  the  differential  equations  completely  by  assuming 
that  the  ratios  of  the  mutual  distances  were  constants.  The  demonstration 
was  repeated  by  Laplace  in  the  Mecanique  Celeste,  vol.  v.,  p.  310.  In  I'Expo- 
sition  du  Systeme  du  Monde  he  remarked  that  if  the  moon  had  been  given  to 
the  earth  by  Providence  to  illuminate  the  night,  as  some  have  maintained,  the 
end  sought  has  been  only  imperfectly  attained;  for,  if  the  moon  were  properly 
started  in  opposition  to  the  sun  it  would  always  remain  there  relatively,  and 
the  whole  earth  would  have  either  the  full  moon  or  the  sun  always  in  view. 
The  demonstration  upon  which  he  based  his  remark  was  made  under  the 
assumption  that  there  was  no  disturbing  force.  If  there  were  disturbing 
forces  the  configuration  would  not  be  preserved  unless  the  solution  were  stable, 
which  it  is  not,  as  was  proved  by  Liouville,  Journal  de  Mathematiques,  vol.  vn., 
1845. 

A  number  of  memoirs  have  appeared  following  more  or  less  closely  along 
the  lines  marked  out  by  Lagrange.  Among  them  may  be  mentioned  one  by 
Radau  in  the  Bulletin  Astronomique,  vol.  in.,  p.  113;  by  Lindstedt  in  the 
Annales  de  VEcole  Normale,  3rd  series,  vol.  i.,  p.  85;  by  Alle^ret  in  the  Journal 
de  Mathematiques,  1875,  p.  277;  by  Bour  in  the  Journal  de  I'Ecole  Poly  technique, 
vol.  xxxvi.;  and  by  Mathieu  in  the  Journal  de  Mathematiques,  1876,  p.  345. 

Jacobi,  without  a  knowledge  of  the  work  of  Lagrange,  reduced  the  general 
Problem  of  Three  Bodies  to  the  seventh  order  in  Crelle's  Journal,  1843,  p.  115 
(Coll.  Works,  vol.  iv.,  p.  478).  It  has  never  been  reduced  further. 

Concerning  the  solutions  of  the  problem  of  more  than  three  bodies  in  which 
the  ratios  of  the  mutual  distances  are  constants  a  number  of  papers  have 
appeared,  among  which  are  one  by  Lehmann-Filhes  in  the  Astronomische 
Nachrichten,  vol.  cxxvu.,  p.  137,  one  by  F.  R.  Moulton  in  The  Transactions  of 
the  American  Mathematical  Society,  vol.  i,,  p.  17,  and  one  by  W.  R.  Longley  in 
Bulletin  of  the  American  Mathematical  Society,  vol.  xin.,  p.  324. 

No  new  periodic  solutions  of  the  problem  of  three  bodies  were  discovered 
after  those  of  Lagrange  until  Hill  developed  his  Lunar  Theory,  The  American 
Journal  of  Mathematics,  vol.  i.  (1878).  These  solutions  of  Hill  are  of  im- 
mensely greater  practical  value  than  those  of  the  Lagrangian  type.  It  should 


320  HISTORICAL    SKETCH. 

be  stated,  however,  that  they  are  not  strictly  periodic  solutions  of  any  actual 
case,  because  a  small  part  of  the  perturbing  action  of  the  sun  was  neglected. 

The  next  important  advance  was  made  by  Poincare  in  a  memoir  in  the 
Bulletin  Astronomique,  vol.  i.,  in  which  he  proved  that  when  the  masses  of  two 
of  the  bodies  are  small  compared  to  that  of  the  third,  there  is  an  infinite 
number  of  sets  of  initial  conditions  for  which  the  motion  is  periodic.  These 
ideas  were  elaborated  and  the  results  extended  in  a  memoir  crowned  with 
the  prize  offered  by  the  late  King  Oscar  of  Sweden.  This  memoir  appeared 
in  Ada  Mathematica,  vol.  xm.  The  methods  employed  by  Poincar6  are 
incomparably  more  profound  and  powerful  than  any  previously  used  in 
Celestial  Mechanics,  and  mark  an  epoch  in  the  development  of  the  science. 
The  work  of  Poincare"  was  recast  and  extended  in  many  directions,  and  pub- 
lished in  three  volumes  entitled,  Les  Methodes  Nouvelles  de  la  Mecanique 
Celeste.  It  is  written  with  admirable  directness  and  clearness,  and  is  given 
in  sufficient  detail  to  make  so  profound  a  work  as  easily  read  as  possible. 

An  important  memoir  on  Periodic  Orbits  by  Sir  George  Darwin  appeared 
in  Acta  Mathematica,  vol.  xxi.  (1899).  In  this  investigation  it  was  assumed 
that  one  of  the  three  masses  is  infinitesimal  and  that  the  finite  masses,  hav- 
ing the  ratio  of  ten  to  one,  revolve  in  circles.  A  large  number  of  periodic 
orbits,  belonging  to  a  number  of  families,  were  discovered  by  numerical  ex- 
periments. The  question  of  their  stability  was  answered  by  using  essen- 
tially the  method  employed  by  Hill  in  his  discussion  of  the  motion  of  the 
lunar  perigee. 

A  considerable  number  of  investigations  in  the  domain  of  periodic  orbits, 
employing  analytical  processes  based  on  the  methods  of  Poincare,  have  been 
published  by  F.  R.  Moulton  and  his  former  students  Daniel  Buchanan,  Thomas 
Buck,  F.  L.  Griffin,  Wm.  R.  Longley,  and  W.  D.  MacMillan.  These  papers 
have  appeared  in  the  Transactions  of  the  American  Mathematical  Society,  the 
Proceedings  of  the  London  Mathematical  Society,  the  Mathematische  Annalen, 
and  the  Proceedings  of  the  Fifth  International  Congress  of  Mathematicians. 
Besides  containing  the  analysis  for  a  great  variety  of  periodic  orbits,  they 
show  the  existence  of  infinite  sets  of  closed  orbits  of  ejection  which  form  the 
boundaries  between  different  classes  of  periodic  orbits.  These  investigations 
are  published  under  the  title  "  Periodic  Orbits  "  as  Publication  161  of  the 
Carnegie  Institution  of  Washington. 


CHAPTER  IX. 

PERTURBATIONS— GEOMETRICAL   CONSIDERATIONS. 

170.  Meaning  of  Perturbations.     It  was  shown  in  chapter  v. 
that  if  two  spherical  bodies  move  under  the  influence  of  their 
mutual  attractions  each  describes  a  conic  section  with  respect  to 
their  center  of  mass  as  a  focus,  and  that  the  path  of  each  body 
with  respect  to  the  other  is  a  conic.     The  converse  theorem  is 
also  true;  that  is,  if  the  law  of  areas  holds  and  if  the  orbit  of  one 
body  is  a  conic  with  respect  to  the  other  as  a  focus,  then  if  the  force 
depends  only  on  the  distance  it  varies  inversely  as  the  square  of 
the  distance  (see  also  Art.  58).     If  there  is  a  resisting  medium, 
or  if  either  of  the  bodies  is  oblate,  or  if  there  is  a  third  body  at- 
tracting the  two  under  consideration,  or  if  there  is  any  force  acting 
upon  the  bodies  other  than  that  of  the  mutual  attractions  of  the 
two  spheres,  their  orbits  will  cease  to  be  exact  conic  sections. 
Suppose  the  coordinates  and  components  of  velocity  are  given  at 
a  definite  instant  tQ;  then,  if  the  conditions  of  the  two-body  problem 
were  precisely  fulfilled,  the  orbits  would  be  definite  conies  in 
which  the  bodies  would  move  so  as  to  fulfill  the  law  of  areas. 
The  differences  between  the  coordinates  and  the  components  of 
velocity  in  the  actual  orbits  and  those  which  the  bodies  would 
have  had  if  the  motion  had  been  undisturbed  are  the  perturbations. 
It  is  necessary  to  include  the  changes  in  the  components  of  velocity 
as  perturbations,  for  the  paths  described  depend  not  only  upon 
the  relative  positions  of  the  bodies  and  the  forces  to  which  they 
are  subject,  but  also  upon  the  relative  velocities  with  which  they 
are  moving. 

Several  methods  of  computing  perturbations  have  been  devised 
depending  upon  the  somewhat  different  points  of  view  which  may 
be  taken.  Of  these  the  two  following  are  the  ones  most  frequently 
used. 

171.  Variation  of    Coordinates.     The    simplest    conception  of 
perturbations  is  that  the  coordinates  are  directly  perturbed.     For 
example,  if  a  planet  is  subject  to  the  attraction  of  another  planet 
the  coordinates  and  components  of  velocity  of  the  former  at  any 
time  t  differ  by  definite  amounts  from  what  they  would  have  been 

22  321 


322  VARIATION    OF   THE    ELEMENTS.  [172 

if  the  sun  had  been  the  only  source  of  attraction,  and  these  differ- 
ences are  computed  by  appropriate  devices.  No  attempt  is  made 
to  get  the  equations  of  the  curve  described,  and  usually  no  general 
information  as  to  what  will  happen  in  the  course  of  a  long  time  is 
secured.  This  method  is  applied  only  to  comets  and  small  planets. 

172.  Variation  of  the  Elements.  This  method  is  variously 
called  the  Variation  of  the  Elements,  the  Variation  of  Parameters, 
and  the  Variation  of  the  Constants  of  Integration.  According  to 
this  conception,  a  body  subject  to  the  law  of  gravitation  is  always 
moving  in  a  conic  section,  but  in  one  which  changes  at  each  instant. 
The  variable  conic  is  tangent  to  the  actual  orbit  at  every  point 


Fig.  43. 

of  it;  and  further,  if  the  body  were  moving  undisturbed  in  any 
one  of  the  tangent  conies  it  would  have  the  same  velocity  at  the 
point  of  tangency  which  it  has  in  the  actual  orbit  at  that  point. 
This  conic  is  said  to  osculate  with  the  actual  orbit  at  the  point  of 
contact.  The  perturbations  are  the  differences  between  the  ele- 
ments of  the  orbit  on  the  start,  and  those  of  the  osculating  conic 
at  any  time.  An  obvious  advantage  of  this  method  is  that  the 
elements  change  very  slowly,  since  in  most  of  the  cases  which 
actually  arise  in  the  solar  system  the  perturbing  forces  are  small. 
But  if  the  perturbations  were  very  large,  as  they  are  in  some  of 
the  multiple  star  systems,  this  method  would  lose  its  relative 
advantages. 


173]  DERIVATION    OF   THE    ELEMENTS.  323 

The  conception  of  perturbations  as  being  variations  of  the 
elements  arises  quite  naturally  in  considering  the  factors  which 
determine  the  elements  of  an  orbit.  It  was  shown  in  chap.  v. 
that  the  initial  positions  of  the  two  bodies  and  the  directions  of 
projection  determine  the  plane  of  the  orbit;  that  the  initial  posi- 
tions and  the  velocities  of  projection  determine  the  length  of  the 
major  axis;  and  that  the  initial  conditions,  including  the  direction 
of  projection  and  the  velocities,  determine  the  eccentricity  and 
the  line  of  the  apsides. 

Suppose  a  body  m  is  projected  from  P0,  Fig.  43,  in  the  direction 
Qo  with  the  velocity  V0.  Suppose  there  are  no  forces  acting  upon 
it  except  the  attraction  of  S',  then,  in  accordance  with  the  results 
of  the  two-body  problem,  it  follows  that  it  will  move  in  a  conic 
section  Co  whose  elements  are  uniquely  determined.  Suppose  that 
when  it  arrives  at  PI  it  becomes  subject  to  an  instantaneous 
impulse  of  intensity /i  in  the  direction  PiQi;  this  position  and  the 
new  velocity  and  direction  of  motion  determine  a  new  conic  Ci  in 
which  the  body  will  move  until  it  is  again  disturbed  by  some 
external  force.  Suppose  it  becomes  subject,  to  the  impulse  /2  in 
the  direction  P2Q2  when  it  arrives  at  P2;  it  will  move  in  the  new 
conic  C2.  This  may  be  supposed  to  continue  indefinitely.  The 
body  will  be  moving  in  conic  sections  which  change  from  time  to 
time  when  it  is  subject  to  the  disturbing  impulses.  Suppose  the 
instantaneous  impulses  become  very  small,  and  that  the  intervals 
of  time  between  them  become  shorter  and  shorter.  The  general 
characteristics  of  the  motion  will  remain  the  same.  At  the  limit 
the  impulses  become  a  continually  disturbing  force,  and  the  orbit 
a  conic  section  which  continually  changes. 

173.  Derivation  of  the  Elements  from  a  Graphical  Construction. 

It  was  shown  in  Art.  89  that  the  major  semi-axis  is  given  by  the 
very  simple  equation 

(1)  V*  - 

where  V  is  the  initial  velocity,  &2'the  Gaussian  constant,  S  +  m 
the  sum  of  the  masses,  r  the  initial  distance  of  the  bodies  from 
each  other,  and  a  the  major  semi-axis.  Suppose  the  major  semi- 
axis  has  been  computed  by  (1) ;  it  will  be  shown  how  the  remaining 
elements  can  be  found  by  the  aid  of  very  simple  geometrical 
constructions.  The  initial  positions  of  S  and  m,  and  the  direction 


324 


RESOLUTION   OF   THE   DISTURBING   FORCE. 


[174 


of  projection  of  m,  determine  the  position  of  the  plane  of  the 
orbit,  and  therefore  &  and  i. 

Suppose  m  is  at  the  point  P  at  the  origin  of  time,  and  that  it  is 
projected  in  the  direction  PQ  with  the  velocity  V.  The  sun  S  is 
at  one  of  the  foci.  It  is  known  from  the  properties  of  conic 
sections  that  the  lines  from  P  to  the  two  foci  make  equal  angles 
with  the  tangent  PQ.  Draw  the  line  PR  making  the  same  angle 
with  the  tangent  that  SP  makes.  Let  ri  represent  the  distance 


from  S  to  P,  and  r2  the  distance  from  P  to  the  second  focus. 
Therefore    rl  +  r2  =  2a;    or,    r2  =  2a  -  rlt    which    defines    the 

SiO 
position  of  Si.     Call  the  mid-point  of  SSi,  0;   then  e  =  --. 

Suppose  S&  is  the  line  of  nodes;  then  the  angle  &SA  =  «,  and 


CO 


The  only  element  remaining  to  be  found  is  the  time  of  perihelion 
passage.  The  angle  ASP,  counted  in  the  direction  of  motion, 
is  v.  The  eccentric  anomaly  is  given  by  the  equation  (Art.  98) 


(2) 


tan 


E_  /r 

2      \1 


—  e 


After  E  has  been  found  the  time  of  perihelion  passage,  T,  is  defined 
by  the  equation  (Art.  93) 


(3) 


n(t  -  T)  =  E  -  e  sin  E. 


174.  Resolution  of  the  Disturbing  Force.  Whatever  may  be 
the  source  of  the  disturbing  force  it  is  convenient,  in  order  to  find 
its  effects  upon  the  elements,  to  resolve  it  into  three  rectangular 
components.  It  is  possible  to  do  this  in  several  ways,  each  having 


175]         DISTURBING   EFFECTS   OF   ORTHOGONAL   COMPONENT.  325 

advantages  for  particular  purposes.  The  one  will  be  adopted 
here  which  on  the  whole  leads  most  simply  to  the  determination 
of  the  manner  in  which  the  elements  vary  when  the  body  under 
consideration  is  subject  to  any  disturbing  force.  It  would  be 
possible  without  much  difficulty  to  derive  from  geometrical  con- 
siderations the  expressions  for  the  rates  of  change  of  the  elements 
for  any  disturbing  forces,  but  the  object  of  this  chapter  is  to 
explain  the  nature  and  causes  of  perturbations  of  various  sorts, 
and  the  attention  will  not  be  divided  by  unnecessary  digressions 
on  methods  of  computation.  This  part  falls  naturally  to  the 
methods  of  analysis,  which  will  be  given  in  the  next  chapter. 

The  disturbing  force  will  be  resolved  into  three  rectangular 
components:  (a)  the  orthogonal  component,*  S,  which  is  per- 
pendicular to  the  plane  of  the  orbit,  and  which  is  taken  positive 
when  directed  toward  the  north  pole  of  the  ecliptic;  (6)  the 
tangential  component,  T,  which  is  in  the  line  of  the  tangent,  and 
which  is  taken  positive  when  it  acts  in  the  direction  of  motion; 
and  (c)  the  normal  component,  'N,  which  is  perpendicular  to  the 
tangent,  and  which  is  taken  positive  when  directed  to  the  interior 
of  the  orbit. 

The  instantaneous  effects  of  these  components  upon  the  various 
elements  will  be  discussed  separately;  and,  unless  it  is  otherwise 
stated,  it  always  must  be  understood  that  the  results  refer  to  the 
way  in  which  the  elements  are  changing  at  given  instants,  and  not 
to  the  cumulative  effects  of  the  disturbing  forces.  Although  the 
effects  of  the  different  components  are  considered  separately,  yet 
when  two  or  more  act  simultaneously  it  is  sometimes  necessary  to 
estimate  somewhat  carefully  the  magnitude  of  their  separate 
perturbations,  in  order  to  determine  the  character  of  their  joint 
effects. 

I.    EFFECTS  OF  THE  COMPONENTS  OF  THE  DISTURBING  FORCE. 
175.  Disturbing   Effects   of   the    Orthogonal   Component.     In 

order  to  fix  the  ideas  and  abbreviate  the  language  it  will  be  sup- 
posed that  the  disturbed  body  is  the  moon  moving  around  the 
earth.  The  perturbations  arising  from  the  disturbing  action  of 
the  sun  are  very  great  and  present  many  features  of  exceptional 
interest.  Besides,  this  is  the  case  which  Newton  treated  by 
methods  essentially  the  same  as  those  employed  here.f  The 

*  A  designation  due  to  Sir  John  Herschel,  Outlines  of  Astronomy,  p.  420. 
t  Prindpia,  Book  i.,  Section  11,  and  Book  m.,  Props  xxn.-xxxv. 


326  DISTURBING   EFFECTS   OF   ORTHOGONAL   COMPONENT.         [175 

character  of  the  perturbations  arising  from  positive  components 
alone  will  be  investigated;  in  every  case  negative  components 
change  the  elements  in  the  opposite  way. 

It  is  at  once  evident  that  the  orthogonal  component  will  not 
change  a,  e,  T,  and  co,  if  co  is  counted  from  a  fixed  line  in  the  plane 
of  the  orbit.  But  the  co  in  ordinary  use  is  counted  from  the 
ascending  node  of  the  orbit;  hence  if  the  negative  of  the  rate  of 
increase  of  ft  be  multiplied  by  cos  i  the  result  will  be  the  rate 
of  increase  of  co  due  to  the  change  in  the  origin  from  which  it  is 
reckoned.  Consequently  it  is  sufficient  to  consider  the  changes 
in  ft  and  i  when  discussing  the  perturbations  due  to  the  orthogonal 
component. 


Fig.  45. 

Let  AB  be  in  the  plane  of  the  ecliptic,  PoQo  in  the  plane  of  the 
undisturbed  orbit,  and  ft0  and  i0  the  corresponding  node  and 
inclination.  Suppose  there  is  an  instantaneous  impulse  PoS0 
when  the  moon  is  at  P0;  it  will  then  move  in  the  direction  PoP\, 
and  the  new  node  and  inclination  will  be  fti  and  i\.  It  is  evident 
at  once  that  i\  >  iQ  and  fti  <  ft0.  Suppose  a  new  instantaneous 
impulse  PiSi  acts  when  the  moon  arrives  at  PI.  The  new  node 
and  inclination  are  ft2  and  iz,  and  it  is  evident  that  i2  <  i\  and 
ft 2  <  fti.  If  Pofti  =  ftiPi,  P0SQ  =  PA,  and  the  velocity  of 
the  moon  at  P0  equals  that  at  PI,  then  i0  =  i2.  The  total  result 
is  a  regression  of  the  node  and  an  unchanged  inclination. 

From  the  corresponding  figure  at  the  descending  node  it  is 
seen  that  a  negative  S  before  node  passage  and  a  symmetri- 
cally opposite  positive  S  after  node  passage  will  produce  the 
same  results  as  those  which  were  found  at  the  ascending  node. 
Therefore,  a  positive  S  causes  the  nodes  to  advance  if  the  moon  is 
in  the  first  or  second  quadrant,  and  to  regress  if  it  is  in  the  third 
or  fourth  quadrant;  and  a  positive  S  causes  the  inclination  to 
increase  if  the  moon  is  in  the  first  or  fourth  quadrant,  and  to 
decrease  if  it  is  in  the  second  or  third  quadrant. 


177]  EFFECTS   OF   TANGENTIAL   COMPONENT.  327 

The  following  quantitative  results  may  be  noted:  The  rate  of 
change  of  both  &  and  i  is  proportional  to  S.  The  rate  of  change 
of  &  is  greater  the  smaller  i\  for  i  =  0  evidently  &  is  not  defined, 
but  in  this  case  in  such  problems  as  the  Lunar  Theory  S  vanishes. 
For  a  given  i  the  rate  of  change  of  &  is  greater  the  nearer  the  point 
at  which  disturbance  occurs  is  to  midway  between  the  two  nodes. 
The  rate  at  which  i  changes  is  greater  the  nearer  the  point  at  which 
the  disturbance  occurs  is  to  a  node. 

176.  Effects  of  the  Tangential  Component  upon  the  Major  Axis. 
Instead  of  deriving  all  the  conclusions  directly  from  geometrical 
constructions,  it  will  be  better  to  make  use  of  some  of  the  simple 
equations  which  have  been  found  in  chapter  v.  If  it  were  desired 
the  theorems  contained  in  these  equations  could  be  derived  from 
geometrical  considerations,  as  was  done  by  Newton  in  the  Prin- 
cipia,  but  this  would  involve  considerable  labor  and  would  add 
nothing  to  the  understanding  of  the  subject. 

The  major  semi-axis  is  given  in  terms  of  the  initial  distance  and 
the  initial  velocity  by  equation  (1);  viz., 


V2  =  k2(E  +  m)     -  -  - 
\r       a 

In  an  elliptic  orbit  a  is  positive ;  hence,  since  a  positive  T  increases 
V2  and  does  not  instantaneously  change  r,  a  positive  T  increases 
the  major  semi-axis  when  the  moon  is  in  any  part  of  its  orbit.  It 
also  follows  from  this  equation  that  a  given  T  is  most  effective  in 
changing  a  when  V  has  its  largest  value,  or  when  the  moon  is  at 
the  perigee,  and  that  the  rate  of  change  is  more  rapid  the  larger  a. 
Expressed  in  terms  of  partial  derivatives,  the  dependence  of  a 
upon  T  is  given  by 

—  -  —  —  2a2V      dV 

'dT~dV'dT~  k2(E  +  m)  ~dT' 

177.  Effects  of  the  Tangential  Component  upon  the  Line  of 
Apsides.  The  tangential  component  increases  or  decreases  the 
speed,  but  does  not  instantaneously  change  the  direction  of 
motion.  The  focus  E  is  of  course  not  changed,  n  is  unchanged, 
and,  according  to  the  results  of  the  last  article,  a'  is  increased. 
Since  r2  =  2a  —  r\  while  the  direction  of  r2  remains  the  same, 
it  follows  that  the  focus  EI  is  thrown  forward  to  EI,  Fig.  46.  The 
line  of  apsides  is  rotated  forward  from  AB  to  A'B'.  Hence  it  is 
easily  seen  that  a  positive  tangential  component  causes  the  line  of 


328 


EFFECTS   OF   TANGENTIAL   COMPONENT. 


[178 


apsides  to  rotate  forward  during  the  first  half  revolution,  and  back- 
ward during  the  second  half  revolution. 

The  instantaneous  effects  are  the  same  for  points  which  are 
symmetrical  with  respect  to  the  major  axis.  When  the  moon  is 
at  K  or  L  the  whole  displacement  of  the  second  focus  is  per- 
pendicular to  the  line  of  apsides,  and  at  these  points  the  rate  of 


Fig.  46. 

rotation  of  the  apsides  is  a  maximum  for  a  given  change  in  the 
major  axis.  But  the  major  axis  is  changed  most  when  the  moon 
is  at  perigee;  therefore  the  place  at  which  the  line  of  the  apsides 
rotates  most  rapidly  is  near  K  and  L  and  between  these  points 
and  the  perigee.  The  rate  of  rotation  of  the  line  of  apsides 
becomes  zero  when  the  moon  is  at  perigee  or  apogee.  It  should 
be  remembered  that  the  whole  problem  is  complicated  by  the 
fact  that  the  magnitude  of  T  depends  upon  the  distances  of  both 
moon  and  sun,  and  these  distances  continually  vary. 

178.  Effects  of  the  Tangential  Component  upon  the  Eccentricity. 

Tjl  Tjl 

The  eccentricity  is  given  by  the  equation  e  =  -=—,   Fig.    46. 

When  the  moon  is  at  the  perigee  EEi  and  2a  are  increased  by  the 
same  amount.  Since  EEi  is  less  than  2a  the  eccentricity  is 
increased  at  this  point.  When  the  moon  is  at  apogee  2a  is  in- 
creased while  EEi  is  decreased  equally,  hence  the  eccentricity  is 
decreased.  Consequently  there  is  some  place  between  perigee 
and  apogee  where  the  eccentricity  is  not  changed,  and  it  is  easy 
to  show  that  this  place  is  at  the  end  of  the  minor  axis.  Let  2Aa 
represent  the  instantaneous  increase  in  2a  when  the  moon  is  at 
C  or  D,  Fig.  47.  Then  r2  will  be  increased  by  the  quantity  2Aa, 

and  EEi  by  A#.     If  6  is  the  angle  CEtE,  cos  e  =  ^  =  2™=e, 


180] 


EFFECTS    OF   NORMAL    COMPONENT. 


329 


and,  moreover,  AE  =  2Aa  cos  6 
EE,  +  AE 


2eAa.     Therefore 
2ae  +  2eAa 


2a  +  2Aa  ~     2a  +  2Aa 


e; 


or,  the  eccentricity  is  unchanged  by  the  tangential  component 
when  the  moon  is  at  an  end  of  the  minor  axis  of  its  orbit. 

The  changes  in  the  time  of  perihelion  passage  depend  upon  the 
changes  in  the  period  and  the  direction  of  the  major  axis,  as  well 
as  on  the  direct  perturbations  of  the  longitude  in  the  orbit.  Since 
the  period  depends  upon  the  major  axis  alone,  whose  changes 


have  been  discussed,  the  foundations  for  an  investigation  of  the 
changes  in  the  time  of  perihelion  passage  have  been  laid,  except 
in  so  far  as  they  are  direct  perturbations  in  longitude;  but  further 
inquiry  into  this  subject  will  be  omitted  because  geometrical 
methods  are  not  well  suited  to  such  an  investigation,  and  because 
the  time  of  perihelion  passage  is  an  element  of  little  interest  in 
the  present  connection. 

179.  Effects  of  the  Normal  Component  upon  the  Major  Axis. 

It  follows  from  (1)  that  the  major  axis  depends  upon  the  speed 
at  a  given  point  and  not  upon  the  direction  of  motion.  Since 
the  normal  component  acts  at  right  angles  to  the  tangent,  it 
does  not  instantaneously  change  the  speed  and,  therefore,  leaves 
the  major  axis  unchanged. 

180.  Effects  of  the  Normal  Component  upon  the  Line  of  Apsides. 

Consider  the  effect  of  an  instantaneous  normal  component  when 
the  moon  is  at  P,  Fig.  48.  Let  PT  represent  the  tangent  to  the  orbit. 
The  effect  of  the  normal  component  will  be  to  change  it  to  PT'. 
Since  the  radii  to  the  two  foci  make  equal  angles  with  the  tangent 


330 


EFFECTS   OF  NORMAL   COMPONENT. 


[180 


the  radius  r2  will  be  changed  to  r2';  and,  since  the  normal  com- 
ponent does  not  affect  the  length  of  the  major  axis,  r2  and  r/ 
will  be  of  equal  length.  Consequently,  when  the  moon  is  in  the 
region  LAK  a  positive  normal  component  will  rotate  the  line  of 
apsides  forward,  and  when  it  is  in  the  region  KBL,  backward.  At 


Fig.  48. 

the  points  K  and  L  the  normal  component  does  not  change  the 
direction  of  the  line  of  apsides. 

In  the  applications  to  the  perturbations  of  the  moon  it  will  be 
important  to  determine  the  relative  effectiveness  of  a  given  normal 
force  in  changing  the  line  of  apsides  when  the  moon  is  at  the  two 
positions  A  and  B.  When  the  moon  is  at  either  of  these  two 
points  the  second  focus  EI  is  displaced  along  the  line  KL.  The 
effectiveness  of  a  force  in  changing  the  direction  of  motion  of  a 
body  is  inversely  proportional  to  the  speed  with  which  it  moves; 
but  by  the  law  of  areas  the  velocities  at  A  and  B  are  inversely 
proportional  to  their  distances  from  E.  Let  EA  and  EB  represent 
the  effectiveness  of  a  given  force  in  changing  the  direction  of 
motion  at  A  and  B  respectively,  and  let  VA  and  VB  represent  the 
velocities  at  the  same  points.  Then 


EA  '  EB  =  VB  :  VA  =  a(l  -  e) 


e). 


The  rotation  of  the  line  of  apsides  is  directly  proportional  to 
the  displacement  of  E\  along  the  line  KL.  The  displacements 
along  KL  are  directly  proportional  to  the  products  of  the  lengths 
of  the  radii  from  A  and  B  to  E\  and  the  angles  through  which  they 
are  rotated.  But  the  angles  are  proportional  to  EA  and  EB,  and 
the  lengths  of  the  radii  to  EI  to  a(l  +  e)  and  a(l  —  e}.  There- 
fore, letting  HA  and  RB  represent  the  rotation  of  the  line  of  apsides 
at  the  two  points,  it  follows  that 


181]  EFFECTS   OF  NORMAL   COMPONENT.  331 

RA  :  KB   =  a(l  +  e)EA  :  a(l  -  e)EB  =  1:1; 

or,  equal  instantaneous  normal  forces  produce  equal,  but  oppositely 
directed,  rotations  of  the  line  of  apsides  when  the  moon  is  at  apogee 
and  at  perigee. 

Suppose  the  forces  act  continuously  over  small  arcs.  Since  the 
linear  velocities  are  inversely  as  the  radii,  the  effectiveness,  in 
changing  the  direction  of  the  line  of  apsides,  of  a  constant  force  acting 
through  a  small  arc  at  A  is  to  that  of  an  equal  force  acting  through 
an  equal  arc  at  B  as  a(l  —  e)  is  to  a(l  +  e).  In  practice  the 
disturbing  forces  are  not  instantaneous  but  act  continuously, 
their  magnitudes  depending  upon  the  positions  of  the  bodies; 
consequently,  unless  the  normal  component  is  smaller  at  apogee 
than  at  perigee  the  average  rotation  of  the  line  of  apsides  due  to  a 
normal  component  always  having  the  same  sign  is  in  the  direction 
of  the  rotation  when  the  moon  is  at  apogee. 

181.  Effects  of  the  Normal  Component  upon  the  Eccentricity. 

If  2a  represents  the  major  axis,  the  eccentricity  is  given  by 

EEl 

e"  -25" 

After  the  action  of  the  normal  component  the  eccentricity  is 


the  major  axis  being  unchanged.  It  is  easily  seen  from  Fig.  48 
that  a  positive  normal  force  decreases  the  eccentricity  during  the  first 
half  revolution  and  increases  it  during  the  second  half,  EE\  being 
less  than  EEi  in  the  first  case,  and  greater  in  the  second.  The 
instantaneous  change  in  the  eccentricity  vanishes  when  the  moon 
is  at  A  or  B. 

It  follows  from  Fig.  48  that  a  given  change  in  the  direction  of  r2 
produces  a  greater  change  in  the  eccentricity  when  the  moon  is 
in  the  second  or  third  quadrant  than  when  the  moon  is  in  a 
corresponding  part  of  the  first  or  fourth  quadrant.  Besides  this, 
the  moon  moves  slower  the  farther  it  is  from  the  earth,  and  conse- 
quently a  given  normal  component  is  more  effective  in  changing 
the  direction  of  motion,  and  therefore  of  r-2,  when  the  moon  is  near 
apogee  than  when  it  is  near  perigee.  Hence  a  given  normal  com- 
ponent causes  greater  changes  in  the  eccentricity  if  the  moon  is  near 
apogee  than  it  does  if  the  moon  is  near  perigee. 


332 


TABLE    OF   RESULTS. 


[182 


182.  Table  of  Results.  The  various  results  obtained  will  be  of 
constant  use  in  the  applications  which  follow,  and  they  will  be 
most  convenient  when  condensed  into  a  table.  The  results  are 
given  for  only  positive  values  of  the  disturbing  components;  for 
negative  components  they  are  the  opposite  in  every  case. 


The  orthogonal  component,  S,  is  positive  when  directed  toward 
the  north  pole  of  the  ecliptic. 

The  tangential  component,  T,  is  positive  when  directed  in  the 
direction  of  motion. 

The  normal  component,  N,  is  positive  when  directed  to  the 
interior  of  the  ellipse. 


Component  .  .  . 

S 

T 

N 

Nodes  

Advance     in     first 

and  second  quad- 
rants; regress,  in 
third  and  fourth 
quadrants. 

0 

0 

Inclination.  .  .  . 

[ncreases  in  first 
and  fourth  quad- 
rants ;  decreases 
in  second  and 
third  quadrants. 

0 

0 

Major  Axis  .  .  . 

0 

Always  increases 

0 

Line  of  Apsides 

No  effect  if  <a  is 
counted  from  a 
fixed  point  rather 
than  from  ft  . 

In  interval  ACS, 
forward  ; 
In  interval  BDA, 
backward 

In  interval  LAK, 
forward; 
In  interval  KBL, 
backward 

Eccentricity.  .  . 

0 

In  interval  DAC, 
increases; 
In  interval  CBD, 
decreases 

In  interval  ACS, 
decreases; 
In  interval  BDA, 
increases 

184] 


PERTURBATIONS  DUE  TO  OBLATE  BODY. 


333 


183.  Disturbing  Effects  of  a  Resisting  Medium.     The  simplest 
disturbance  of  elliptic  motion  is  that  arising  from  a  resisting 
medium.     The   only   disturbing   force   is   a  negative   tangential 
component,  which  has  the  same  magnitude  for  points  symmetri- 
cally situated  with  respect  to  the  major  axis.     Therefore,  it  is 
seen  from  the  Table  that:   (1)  &  and  i  are  unchanged;  (2)  a  is 
continually  decreased;  (3)  the  line  of  apsides  undergoes  periodic 
variations,  rotating  backward  during  the  first  half  revolution, 
and  rotating  forward  equally  during  the  second  half;   (4)   the 
eccentricity  decreases  while  the  body  moves  through  the  interval 
DAC,  and  increases  during  the  remainder  of  the  revolution.     It 
takes  the  body  longer  to  move  through  the  arc  CBD  than  through 
DAC\  but,  on  the  other  hand,  if  the  resistance  depends  on  a  high 
power  of  the  velocity,  as  experiment  shows  it  does  for  high  veloci- 
ties, the  change  is  much  greater  at  perigee  than  at  apogee,  and 
the  whole  effect  in  a  revolution  is  a  decrease  in  the  eccentricity. 
The  application  of  these  results  to  a  comet,  planet,  or  satellite 
resisted  by  meteoric  matter,  or  possibly  the  ether,  is  evident. 

184.  Perturbations  Arising  from   Oblateness   of  the   Central 
Body.     Consider  the  case  of  a  satellite  revolving  around  an  oblate 
planet  in  the  plane  of  its  equator.     It  was  shown  in  equations 
(30),  p.  122,  that  the  attraction  under  these  circumstances  is  always 
greater  than  that  of  a  concentric  sphere  of  equal  mass,  but  that 


D 
Fig.  50. 

the  two  attractions  approach  equality  as  the  satellite  recedes. 
The  excess  of  the  attraction  of  the  spheroid  over  that  of  an  equal 
sphere  will  be  considered  as  being  the  disturbing  force,  which, 
it  will  be  observed,  acts  in  the  line  of  the  radius  vector  and  is 
always  directed  toward  the  planet.  Therefore  the  normal  com- 


334  PERTURBATIONS  DUE   TO    OBLATE   BODY.  [184 

ponent  is  always  positive,  and  is  equal  in  value  at  points  which 
are  symmetrically  situated  with  respect  to  the  major  axis.  If  the 
eccentricity  of  the  orbit  is  not  large  the  tangential  component  is 
relatively  small,  being  negative  in  the  interval  ACB,  and  positive 
in  BDA. 

(a)  Effect  upon  the  period.  This  is  most  easily  seen  when  the 
orbit  is  a  circle.  The  attraction  will  be  constant  and  greater 
than  it  would  be  if  the  planet  were  a  sphere.  This  is  equivalent 
to  increasing  k2,  the  acceleration  per  unit  mass  at  unit  distance; 
therefore  it  is  seen  from  the  equation 

P  = 


that  for  a  given  orbit  the  period  will  be  shorter,  and  for  a  given 
period  the  distance  greater,  than  it  would  be  if  the  planet  were  a 
sphere. 

(b)  Effects  upon  the  elements.  On  referring  to  the  Table,  it  is 
seen  that:  (1)  &  and  i  are  unchanged;  (2)  a  decreases  and  in- 
creases equally  in  a  revolution;  (3)  the  line  of  apsides  rotates 
forward  during  a  little  more  than  half  a  revolution,  and  that  while 
the  disturbing  force  is  of  greatest  intensity ;  and  (4)  the  eccentricity 
is  changed  equally  in  opposite  directions  in  a  whole  revolution. 
That  is,  &  and  i  are  absolutely  unchanged;  a  and  e  undergo  periodic 
variations  which  complete  their  period  in  a  revolution;  and  the  line 
of  apsides  oscillates,  but  advances  on  the  whole. 

The  effects  will  be  the  greater  the  more  oblate  the  planet  and 
the  nearer  the  satellite.  The  oblateness  of  the  earth  is  so  small 
that  it  has  very  little  effect  in  rotating  the  moon's  line  of  apsides. 
The  most  striking  example  of  perturbations  of  this  sort  in  the 
solar  system  is  in  the  orbit  of  the  Fifth  Satellite  of  Jupiter.  This 
planet  is  so  oblate  and  the  satellite's  orbit  is  so  small  that  its 
line  of  apsides  advances  about  900°  in  a  year. 


PROBLEMS. 


335 


XXIII.     PROBLEMS. 

1.  A  body  subject  to  no  forces  moves  in  a  straight  line  with  uniform  speed. 
The  elements  of  this  orbit  are  the  constants  which  define  the  position  of  the 
line,  viz.,  the  speed,  the  direction  of  motion  in  the  line,  and  the  position  of 
the  body  at  the  time  T.     Show  that  they  can  be  expressed  in  terms  of  six 
independent  constants,  and  that  it  is  permissible  in  the  problem  of  two  bodies 
to  regard  one  body  as  always  moving  with  respect  to  the  other  in  a  straight 
line  whose  position  continually  changes.     Find  the  expression  of  these  line 
elements  in  terms  of  the  time  in  the  case  of  elliptic  motion. 

2.  Show  from  general  considerations  based  on  problem  1  that  the  methods 
of  the  variation  of  coordinates  and  the  variation  of  parameters  are  essentially 
the  same,  differing  only  in  the  variables  used  in  denning  the  coordinates  and 
velocities  of  the  bodies. 

3.  Suppose  the  sun  moves  through  space  in  the  line  L,  orthogonal  to  the 
plane  II .     Take  n  as  the  fundamental  plane  of  reference.     Let  the  point 
where  the  planet  Pi  passes  through  the  plane  n  in  the  direction  of  the  motion 
of  the  sun  be  the  ascending  node,  and,  beginning  at  this  point,  divide  the 
orbit  into  quadrants  with  respect  to  the  sun  as  center.     Suppose  the  ether 
and  scattered  meteoric  matter  slightly  retard  the  sun  and  the  planets,  but 
neglect  the  retardation  arising  from  the  motion  of  the  planets  in  their  orbits 
around  the  sun. 

(a)  If  the  resistance  is  proportional  to  the  masses  of  the  respective  bodies, 
show  that  the  nodes  and  inclinations  of  their  orbits  are  unchanged. 

(6)  Let  a  and  R  represent  the  density  and  radius  of  the  sun,  and  o-;  and  Ri 
the  corresponding  quantities  for  the  planet  P».  Then,  if  the  resistance  is 
proportional  to  the  surfaces  of  the  respective  bodies,  show  that  with  respect 
to  the  plane  II  the  inclination  and  line  of  nodes  undergo  the  following  vari- 
ations: 

(1)     If  (nRi  <  aR. 


Quadrant 

1 

2 

3 

4 

Inclination  

decreases 

increases 

increases 

decreases 

Line  of  nodes  

regresses 

regresses 

advances 

advances 

(2)     If  info  >  <rR. 


Quadrant 

1 

2 

3 

4 

Inclination 

increases 

decreases 

decreases 

Line  of  nodes  

advances 

advances 

regresses 

regresses 

336  PROBLEMS. 

(c)  If  the  orbits  were  circles  the  various  changes  in  both  cases  would 
exactly  balance  each  other  in  a  whole  revolution.     How  must  the  lines  of 
apsides  in  the  two  cases  lie  with  respect  to  the  line  of  nodes  in  order  that,  for 
a  few  revolutions,  (1)  the  inclination  shall  decrease  the  fastest,  and  (2)  the 
line  of  nodes  advance  the  fastest? 

(d)  Is  it  possible  to  make  the  relation  of  the  line  of  apsides  to  the  line 
of  nodes  such  that,  for  a  few  revolutions,  the  inclination  shall  decrease  and 
the  line  of  nodes  advance? 

(e)  If  the  line  of  apsides  remains  fixed  in  the  plane  of  the  orbit  is  it  possible 
for  the  line  of  nodes  to  rotate  indefinitely  in  one  direction? 

4.  Suppose  the  orbit  of  a  comet  passes  near  Jupiter's  orbit  at  one  of  its 
nodes;  under  what  conditions  will  the  inclination  of  the  orbit  of  the  comet 
be  decreased?     Show  that  if  the  major  axis  remains  constant  while  the  in- 
clination is  decreased  the  eccentricity  is  increased.     (Use  Art.  159.) 

5.  What  is  the  effect  of  the  gradual  accretion  of  meteoric  matter  by  a 
planet  upon  the  major  axis  of  its  orbit? 

6.  Consider  two  viscous  bodies  revolving  around  their  common  center  of 
mass,  and  rotating  in  the  same  direction  with  periods  less  than  their  period 
of  revolution.     They  will  generate  tides  in  each  other  which  will  lag.     The 
tidal  protuberances  of  each  body  will  exert  a  positive  tangential  and  a  positive 
normal  component  on  the  other,  these  components  being  greater  the  nearer 
the  bodies  are  together.     Moreover,  the  rotation  of  each  body  will  be  retarded 
by  the  action  of  the  other  on  its  protuberances.     Suppose  the  bodies  are 
initially  near  each  other  and  that  their  orbits  are  slightly  elliptic;  follow  out 
the  evolution  of  all  of  the  elements  of  their  orbits. 


185]  DISTURBING   EFFECTS   OF   A   THIRD    BODY.  337 


II.     THE  LUNAR  THEORY. 

185.  Geometrical  Resolution  of  the  Disturbing  Effects  of  a 
Third  Body.  The  problem  of  the  disturbance  by  a  third  body 
is  much  more  difficult  than  those  treated  in  Arts.  183  and  184, 
because  the  disturbing  force  varies  in  a  very  complicated  manner. 


Fig.  51. 

Suppose  the  three  bodies  are  S,  E,  and  m,  and  consider  S  as 
disturbing  the  motion  of  m  around  E.  Two  positions  of  m  are 
shown  at  mi  and  w2,  and  all  the  statements  which  are  made  apply 
for  both  subscripts.  Let  EN  represent  in  magnitude  and  direction 
the  acceleration  of  S  on  E.  The  order  of  the  letters  indicates  the 
direction  of  the  vector  representing  the  force,  and  the  magnitude 
of  the  vector  depends  upon  the  units  employed.  In  the  same 
units  let  mK  represent  in  direction  and  amount  the  acceleration 
of  S  on  m.  The  vector  m\K\  is  greater  than  EN  because  m\S  is 
less  than  ES,  and  mzKz  is  less  than  EN  because  w2$  is  greater 
than  ES.  By  the  law  of  gravitation  they  are  proportional  to  the 
inverse  squares  of  the  respective  distances. 

Now  resolve  mK  into  two  components,  mL  and  mP,  such  that 
mL  shall  be  equal  and  parallel  to  EN.  Since  mL  and  EN  are  equal 
and  parallel  these  components  will  not  disturb  the  relative  posi- 
tions of  E  and  m.  Therefore  the  disturbing  acceleration  is  mP. 

One  important  result  is  evident  from  Fig.  51,  viz.,  that  the 
disturbing  acceleration  is  always  toward  the  line  joining  E  and  S} 
or  toward  this  line  extended  beyond  E  in  the  direction  opposite 
to  S  when  mS  is  greater  than  ES.  Similar  considerations  applied 
to  movable  particles  on  the  surface  of  the  earth  show  why  there 
tends  to  be  a  tide  both  on  the  side  of  the  earth  toward  the  moon, 
and  also  on  the  opposite  side. 
23 


338 


DISTURBING   EFFECTS    OF   A   THIRD   BODY. 


[186 


186.  Analytical  Resolution  of  the  Disturbing  Effects  of  a  Third 
Body.  Take  a  system  of  rectangular  axes  with  the  origin  at  the 
earth  and  with  the  xy-pl&ne  as  the  plane  of  the  ecliptic.  Let 
(x,  y,  z)  and  (X,  Y,  0)  be  the  coordinates  of  the  moon  and  sun 
respectively  referred  to  this  system.  Let  r,  p,  and  R  represent 
the  distances  Em,  mS,  and  ES  respectively.  Let  Fx,  Fy,  and  Fz 
represent  the  components  of  the  disturbing  acceleration  parallel 
to  the  x,  y,  and  z-axes  respectively.  It  follows  from  equations 
(24)  of  chapter  vn.,  p.  272,  that  in  the  present  notation 


Fig.  52. 


(4)    < 


By 


R3 


In  order  to  get  the  components  of  the  disturbing  acceleration  in 
any  other  directions  it  is  sufficient  to  project  these  three  com- 
ponents on  lines  having  those  directions  and  to  take  the  respective 
sums. 

Let  Fr  represent  the  component  of  the  disturbing  acceleration 
in  the  direction  of  the  radius  vector  r;  let  Fv  represent  the  com- 
ponent in  a  line  perpendicular  to  r  in  the  plane  of  motion  of  m; 
and  let  Fy  represent  the  component  which  is  perpendicular  to 


186] 


DISTURBING    EFFECTS    OF   A   THIRD    BODY. 


339 


both  Fr  and  Fv.  The  component  Fr  will  be  taken  as  positive  when 
it  is  directed  from  E\  the  component  Fv  will  be  taken  positive  when 
it  makes  with  the  direction  of  motion  an  angle  less  than  90°;  and 
the  component  F&  will  be  taken  positive  when  it  is  directed  to 
the  hemisphere  which  contains  the  positive  end  of  the  z-axis. 
The  expression  for  Fr  is 

Fr  =  Fx  cos  (xEm)  +  Fy  cos  (yEm)  +  Fz  cos  (zEm). 

The  expression  for  Fv  can  be  obtained  from  this  one  by  replacing 
the  angle  &Em  by  &>Em  +  90°,  because  r  will  have  the  direction 
of  the  tangent  at  m  after  the  body  has  moved  forward  90°  in  its 
orbit.  The  expression  for  FN  can  be  conveniently  obtained  by 
first  projecting  Fx  and  Fy  on  a  line  in  the  xy-pl&ne  which  is  per- 
pendicular to  l£ft,  then  projecting  this  result  on  the  line  perpen- 
dicular to  the  plane  &>Em,  and  projecting  Fz  directly  on  the 
same  final  line.  Let  the  angle  &>Em  be  represented  by  u;  then 
it  is  found  from  Fig.  52  by  spherical  trigonometry  that 

Fr  =  +  Fx[cos  u  cos  ft  —  sin  u  sin  ft  cos  i] 
+  Ftf[cos  u  sin  ft  +  sin  u  cos  ft  cos  i] 
+  Fz  sin  u  sin  i, 

(5)  -j  Fv  =  +  Fx[—  sin  u  cos  ft  —  cos  u  sin  ft  cos  i] 

+  Fy[—  sin  u  sin  ft  +  cos  u  cos  ft  cos  i] 
-\-  Fz  cos  u  sin  i, 
=  +  Fx  sin  ft  sin  i  —  Fy  cos  ft  sin  i  +  Fz  cos  i. 

Let  U  represent  the  angle  ft#*S;  then,  since  the  sun  moves  in 
the  :n/-plane, 

x  =  r[cos  u  cos  ft  —  sin  u  sin  ft  cos  i], 
y  =  r[cos  u  sin  ft  +  sin  u  cos  ft  cos  i], 

z  =  r  sin  u  sin  i\ 

(6)  \ 

X  =  R[cos  U  cos  ft  —  sin  U  sin  ft], 

F  =  R[cos  U  sin  ft  +  sin  U  cos  ft], 
Z  =  0. 


On  substituting  the  expressions  for  FT, 
use  of  (6),  and  reducing,  it  is  found  that 


and  Fz  in  (5),  making 


340 


DISTURBING   EFFECTS    OF   A   THIRD    BODY. 


[186 


(7)    - 


Fr   =    k*S        - 


R    cos  U  cos  u 


r 


+  sin  U  sin  u  cos  i\     —  —  -^  \\ , 


cos  17  sin  u 
+  sin  £7  cos  u  cos  i 
0  —  R  sin  C7  sin  t     -=  —  •=- 


•][?-*]}• 


The  geometry  of  equations  (7)  is  important  for  a  complete 
understanding  of  the  problem.  Consider  a  system  of  axes  with 
origin  at  E,  one  axis  directed  toward  m,  another  at  right  angles 
to  it  and  90°  forward  in  the  plane  of  the  orbit  of  m,  and  the  third 
perpendicular  to  the  other  two.  Then  it  follows  from  the  figure 

that  the  coefficients  of  kzSR    -5  —  j^     in  (7)  are  respectively  the 

cosines  of  the  angles  between  these  axes  and  the  line  ES.  There- 
fore Fv  vanishes  if  the  line  through  E  parallel  to  the  perpendicular 
to  the  radius  is  also  perpendicular  to  ES,  and  FN  vanishes  if  m 
is  in  the  plane  of  the  orbit  of  S.  They  both  vanish  also  if  r  =  p} 

k2S 
and  in  this  case  Fr  becomes  simply ^  . 

Let  \f/  represent  the  angle  between  r  and  R',  then 


(8) 


Therefore  the  expression  for  Fr  becomes 

(9)  ^r 


p2  =  122  +  r2  -  2/^r  cos  ^, 

2r  ,    r2 


Consequently  Fr  vanishes,  if  the  terms  of  higher  order  are  ne- 
glected, when 


(10) 


i; 


+  3  cos  2\f/  =  0,  whence 

=  54°  44/f     125°  16',     234°  44',     305°  16;. 

Now  consider  the  problem  of  finding  the  tangential  and  normal 
components  of  the  disturbing  acceleration.     Let  P  represent  a 


186] 


DISTURBING   EFFECTS    OF   A   THIRD    BODY. 


341 


general  point  in  the  orbit,  Fig.  53.  Let  PT  be  the  tangent  at  P 
and  PN  the  perpendicular  to  it.  It  follows  from  the  elementary 
properties  of  ellipses  that  PN  bisects  the  angle  between  n  ancj  r2. 


Fig.  53. 

Then  the  tangential  and  normal  components  of  the  disturbing 
acceleration  are  expressed  in  terms  of  Fr  and  Fv  by 


(11) 


T  =  +  Fr  sin  6  +  Fv  cos  e, 

N  =  -  Fr  cos  0  +  Fv  sin  0. 


In  order  to  complete  the  expressions  for  T  and  N  the  factors 
sin  6  and  cos  0  must  be  expressed  in  terms  of  v.  It  follows  from 
the  geometrical  properties  of  the  ellipse  and  from  the  triangle 
t  that 


1  +  e  cos  v 
+  r2  =  2a, 

+  r22  -  2rir2  cos  20  =  4a2e2. 


When  7*1  and  r2  are  eliminated  from  these  three  equations,  it  is 

found  that 

e  sin  v 


.  .  . 
sin  6 


. 
VI  +  e2  +  2e  cos 

1  +  e  cos  i; 


Therefore 


(12) 


T  = 


S 


e  sin  # 
Vl  +  e2  +  2e  cos  v 


l  +  e2  +  2e  cos  v 


(1  +  e  cos  t;) 


Vl  +  e2  +  2e  cos  v 

e  sin  v 

Vl  +  e2  +  2e  cos  v 


Fv. 


342 


PERTURBATIONS   OF   THE   NODE. 


[187 


On  making  use  of  (7)  and  the  relation  u  =  co  +  vt  the  final 
expressions  for  the  tangential  and  normal  components  of  the 
disturbing  acceleration  become 


(13) 


T  = 


{—  e  sin  v  -= 
P3 


Vl  +  e2  +  26  cos 

+     —  cos  U  (sin  u  +  e  sin  «) 

+  sin  £7  cos  i  (cos  w  +  e  cos  w)  \R    —  —  ^     | , 


Vl  +  e2  +  26  cos  v 

—    cos  U  (cos  w  +  e  cos  w) 

+  sin  U  cos  i  (sin  u  +  e  sin  co)  \R  \  -^  —  ^     | . 

All  the  circumstances  of  the  variation  of  T  and  TV  can  be  inferred 
from  these  equations. 

187.  Perturbations  of  the  Node.     By  definition,  the  orthogonal 
component  S  is  identical  with  F# ;  therefore  by  the  last  of  (7) 


(14)      Orthog.  Comp.  =  S  =  -  k2SR  sin  U  sin  i  \  -3  -  -^     . 

The  sign  of  the  right  member  depends  upon  the  signs  of  sin  U  and 
~s  ~~  #s     '  both  of  which  can  be  either  positive  or  negative. 

In  order  to  determine  which  sign  prevails  in  the  long  run  so  as 
to  find  whether  on  the  whole  there  is  an  advance  or  retrogression 
of  the  line  of  nodes,  it  is  necessary  to  expand  the  last  factor  of  (14). 
On  making  use  of  the  last  equation  of  (8),  it  is  found  that 


(15) 


S  = ™p-  sin  U  sin  i  cos  ^  +  •  •  • 

3/b2 ST 
•• o?~  sm  U  sm  ^Icos  U  cos  u  +  sin  U  sin  u  cos  i]  +  •  •  • , 


where  the  S  in  the  right  member  represents  the  mass  of  the  sun. 

The  angular  velocity  of  the  sun  in  its  orbit  is  slow  compared  to 

that  of  the  moon;  hence,  in  order  to  simplify  the  discussion,  it 

may  be  supposed  to  stand  still  while  the  moon  makes  a  single 


188]  PERTURBATIONS   OF   THE   INCLINATION.  343 

revolution.  Since  the  periods  of  the  moon  and  sun  have  no  simple 
relation  the  values  of  sin  U  and  cos  U  in  the  long  run  will  be  as 
often  decreasing  as  increasing,  and  hence  the  assumption  will 
cause  no  important  error. 

Suppose  S  is  broken  up  into  the  sum  of  two  parts,  Si  and  $2, 
where 


(16) 


a  .     .    .     TT        TT 

Si  =  --  D-f"  sm  l  sm  "  cos  "  cos  u> 


sin  i  cos  i  sin2  U  sin  u. 


p 


In  order  to  get  the  greatest  degree  of  simplicity  suppose  the  orbit 
of  the  moon  is  a  circle  so  that  r  is  a  constant  and  u  =  nt.  Suppose 
U  has  a  definite  value  and  consider  the  effects  of  Si  during  a  revo- 
lution of  the  moon,  starting  with  the  ascending  node.  It  follows 
from  the  table  of  Art.  182  that  the  effects  of  Si  in  the  first  and 
second  quadrants  are  equal  and  opposite  because  cos  u  has  equal 
numerical  values  and  opposite  signs  in  the  two  quadrants.  It 
is  the  same  in  the  third  and  fourth  quadrants.  Therefore  Si 
produces  only  periodic  perturbations  in  the  line  of  nodes. 

Now  consider  the  effects  of  S&  In  the  first  half  revolution, 
starting  with  the  node,  $2  is  negative  because  sin  u  is  positive 
and  all  the  other  factors  are  positive.  In  the  second  half  revo- 
lution 82  is  positive  because  sin  u  is  negative.  Therefore,  it 
follows  from  the  table  of  Art.  182  that  82  causes  a  continuous,  but 
irregular,  regression  (except  when  it  is  temporarily  zero)  of  the  line 
of  nodes.  The  complete  motion  of  the  line  of  nodes  is  the  resultant 
of  the  periodic  oscillations  due  to  Si  and  the  periodic  and  con- 
tinuous changes  produced  by  S2. 

The  period  of  revolution  of  the  moon's  line  of  nodes  is  about 
nineteen  years.  Since  eclipses  of  the  sun  and  moon  can  occur 
only  when  the  sun  is  near  a  node  of  the  moon's  orbit,  the  times  of 
the  year  at  which  they  take  place  are  earlier  year  after  year,  the 
cycle  being  completed  in  about  nineteen  years. 

188.  Perturbations  of  the  Inclination.  The  expression  for  the 
orthogonal  component  is  given  in  (15),  which  may  again  be  broken 
up  into  the  two  parts  Si  and  $2.  It  follows  from  the  table  of  Art. 
182  that  a  positive  S  increases  the  inclination  in  the  first  and  fourth 
quadrants  and  decreases  it  in  the  second  and  third  quadrants. 

Consider  the  effects  of  Si.     If  sin  U  cos  U  is  positive  the  effect 


344  PEECESSION   OF   THE   EQUINOXES.       NUTATION.  [189 

in  each  quadrant  is  to  decrease  the  inclination.  But  this  case 
can  be  paired  with  that  in  which  sin  U  cos  U  is  negative  and  of 
equal  numerical  value.  Since  all  possible  situations  can  be  paired 
in  this  way,  Si  produces  only  periodic  changes  in  the  inclination. 

The  case  of  $2  is  even  simpler  than  that  of  Si.  Since  sin  u  is 
positive  in  the  first  two  quadrants,  the  effect  in  the  second  quad- 
rant offsets  that  in  the  first.  Similarly,  the  effects  in  the  third 
and  fourth  quadrants  mutually  destroy  each  other.  Therefore  the 
inclination  undergoes  only  periodic  variations. 

Some  things  have  been  neglected  in  this  discussion  to  which 
attention  should  be  called.  No  account  has  been  taken  of  the 
eccentricities  of  the  orbits  of  the  moon  and  earth.  When  they 
are  included  the  terms  do  not  completely  destroy  one  another  in 
the  simple  fashion  which  has  been  described.  Moreover,  each 
perturbation  has  been  considered  independently  of  all  other  ones. 
As  a  matter  of  fact,  each  one  depends  en  all  the  others.  For 
example,  if  the  node  changes,  the  effects  on  the  inclination  are 
different  from  what  they  would  otherwise  have  been,  and  con- 
versely. It  is  clear  that  a  very  refined  analysis  is  necessary  in 
order  to  get  accurate  numerical  results.  But  this  does  not  mean 
that  common-sense  geometrical  and  physical  considerations  are 
not  of  the  highest  importance,  especially  in  first  penetrating 
unexplored  fields. 

189.  Precession  of  the  Equinoxes.  Nutation.  Suppose  the 
largest  sphere  possible  is  cut  out  of  the  earth  leaving  an  equatorial 
ring.  Every  particle  in  this  ring  may  be  considered  as  being  a 
small  satellite;  then,  from  the  principles  explained  in  Arts.  185 
and  186,  the  attractions  of  the  moon  and  sun  will  exercise  dis- 
turbing accelerations  upon  them  which  will  tend  to  shift  them 
with  respect  to  the  spherical  core.  But  the  particles  of  the  ring 
are  fastened  to  the  solid  earth  so  that  it  partakes  of  any  dis- 
turbance to  which  they  may  be  subject.  Since  their  combined 
mass  is  very  small  compared  to  that  of  the  spherical  body  within 
them,  and  since  the  disturbing  forces  are  very  slight,  the  changes 
in  the  motion  of  the  earth  will  take  place  very  slowly. 

From  the  results  of  the  last  article  it  follows  that  the  nodes  of 
the  orbit  of  every  particle  will  have  a  tendency  to  regress  on  the 
plane  of  the  disturbing  body.  The  angle  between  the  plane  of 
the  moon's  orbit  and  that  of  the  ecliptic  may  be  neglected  for  the 
moment  as  it  is  small' compared  to  the  inclination  of  the  earth's 


190]  RESOLUTION   OF  DISTURBING  ACCELERATION.  345 

equator.  They  communicate  this  tendency  to  the  whole  earth 
so  that  the  plane  of  the  earth's  equator  turns  in  the  retrograde 
direction  on  the  plane  of  the  ecliptic.  On  the  other  hand,  it  follows 
from  the  symmetry  of  the  figure  with  respect  to  the  nodes  of  the 
orbits  of  the  particles  of  the  equatorial  ring  that  there  will  be  no 
change  in  the  inclination  of  the  plane  of  the  equator  to  that  of 
the  ecliptic  or  the  moon's  orbit.  The  mass  moved  is  so  great, 
and  the  forces  acting  are  so  small,  that  this  retrograde  motion, 
called  the  precession  of  the  equinoxes,  amounts  to  only  about 
50".2  annually;  or,  the  plane  of  the  earth's  equator  makes  a  revo- 
lution in  about  26,000  years. 

The  moon  is  very  near  to  the  earth  compared  to  the  sun,  and  the 
orthogonal  component  arising  from  its  attraction  is  greater  than 
that  coming  from  the  sun's  attraction.  The  main  regression  is, 
therefore,  on  the  moon's  orbit,  which  is  inclined  to  the  ecliptic 
about  5°  9'.  Since  the  line  of  the  moon's  nodes  makes  a  revo- 
lution in  about  19  years,  the  plane  with  respect  to  which  the 
equator  regresses  performs  a  revolution  in  the  same  time.  This 
produces  a  slight  nodding  in  the  motion  of  the  pole  of  the  equator 
around  the  pole  of  the  ecliptic,  and  is  called  nutation. 

The  quantitative  agreement  between  theory  and  observation  of 
the  rate  of  precession  proves  that  the  equatorial  bulge  is  solidly 
attached  to  the  remainder  of  the  earth.  If  the  earth  were  a 
relatively  thin  solid  crust  floating  on  a  liquid  interior,  as  was  once 
supposed,  it  would  probably  slide  somewhat  on  the  interior  and 
give  a  more  rapid  precession. 

190.  Resolution  of  the  Disturbing  Acceleration  in  the  Plane  of 
Motion.  It  follows  from  the  table  of  Art.  182  that  the  orthogonal 
component  does  not  produce  perturbations  in  the  major  axis, 
longitude  of  perigee,  and  eccentricity,  except  indirectly  as  it 
shifts  the  line  of  nodes  from  which  the  longitude  of  the  perigee  is 
counted.  Consequently  an  idea  of  the  way  these  elements  are 
perturbed  can  be  obtained  even  if  the  inclination,  with  which  the 
orthogonal  component  vanishes,  is  supposed  to  be  zero.  But  it 
must  be  remembered  the  results  obtained  under  these  restrictions 
are  not  rigorous  because  T  and  N  depend  on  the  inclination.  But 
the  approximation  is  fully  justified  because  it  results  in  great 
simplifications  which  aid  correspondingly  in  understanding  the 
subject. 

On  taking  i  =  0  equations  (13)  become 


346  RESOLUTION    OF   DISTURBING   ACCELERATION.  [190 


(17) 


T   = 


N  = 


1  —  e  sin  v  -= 
P3 


VI  +  e2  +  2e  cos 

-fl[sin  (t*  -  CO  +  esin  («  -  CO]  [?-gi]}i 


Vl  +  e2  -h  2e  cos  v 


Tangential  Component. 


When  i  equals  zero  ^  =  u  —  U,  and  on  using  the  last  equation 
of  (8),  it  is  found  that 

7-9  Of  f 

T  = 


(18) 


r    . 


—  Seisin  (o>  —  U)  cos  (u  —  U) 


In  the  orbit  of  the  moon  e  is  approximately  equal  to  ^ 
consequently  a  good  idea  of  the  numerical  magnitudes  of  T  and  N 
and  the  circumstances  under  which  they  change  sign  can  be 


191] 


PERTURBATION    OF   THE   MAJOR   AXIS. 


347 


obtained  by  neglecting  those  terms  which  have  e  as  a  factor.  If 
these  terms  are  neglected  it  is  found  that  T  vanishes  at  u  —  U  =  0 

Q 

- ,  TT,  and  --  ;  it  is  negative  in  the  first  and  third  quadrants,  and 

&  z 

positive  in  the  second  and  fourth  quadrants.  Under  the  same 
circumstances  N  vanishes  at  54°  44',  125°  16',  234°  44',  and 
305°  16';  it  is  negative  from  -  54°  44'  to  +  54°  44'  and  from 
125°  16'  to  234°  44',  and  is  positive  from  54°  44'  to  125°  16'  and 
from  234°  44'  to  305°  16'.  If  the  terms  depending  on  e  and  the 

Normal  Component. 
m. 


Fig.  55. 

higher  terms  in  the  expansion  of  p"3  are  retained,  the  points 
where  T  and  N  vanish  are  in  general  slightly  different  from  those 
which  have  been  found,  but  the  differences  are  not  important  in 
a  qualitative  discussion  whose  aim  is  simply  to  exhibit  the  general 
characteristics  of  the  results. 

The  signs  of  T  and  N  for  the  moon  in  different  parts  of  its  orbit 
are  shown  in  Figs.  54  and  55. 

191.  Perturbations  of  the  Major  Axis.  If  the  perigee  were 
at  mi  or  m^  the  tangential  component,  which  alone  changes  a, 
would  be  equal  and  of  opposite  sign  at  points  symmetrically 
situated  with  respect  to  the  major  axis.  In  this  case  a  would  be 
unchanged  at  the  end  of  a  complete  revolution.  But  this  con- 
dition of  affairs  is  only  realized  instantaneously,  for  the  disturbing 
body  S  is  moving  in  its  orbit;  yet,  in  a  very  large  number  of  revo- 
lutions, when  the  periods  are  incommensurable,  an  equal  number 
of  equal  positive  and  negative  tangential  components  will  have 


348  PERTURBATION    OF   THE    PERIOD.  [192 

exerted  a  disturbing  influence.     The  result  is  that  in  the  long 
run  a  is  unchanged,  although  it  undergoes  periodic  variations. 

192.  Perturbation  of  the  Period.  The  normal  component  is 
not  only  negative  more  than  half  a  revolution,  but  the  negative 
values  are  greater  numerically  than  the  positive  ones.  If  the  terms 
involving  e  are  neglected,  it-Js^seen  from  the  second  equation  of 
(18)  that  thejgeatest  pSswevalue  of  N  is  twice  its  numerically 
greatest  iregSfeTvalue.  One  effect  of  the  whole  result  is  equiva- 
lent to  a  diminution,  on  the  average,  of  the  attraction  of  E  for  m; 
that  is,  to  a  diminution  of  k2,  the  acceleration  at  unit  distance. 
The  relation  of  the  period  to  the  intensity  of  the  attraction  and 
the  major  axis  is  (Art.  89) 


Hence,  for  a  given  distance,  P  is  increased  if  k  is  decreased.  In 
this  manner  the  sun's  disturbing  effect  upon  the  orbit  of  the  moon 
increases  the  length  of  the  month  by  more  than  an  hour.  (Com- 
pare Art.  184  (a).) 

193.  The  Annual  Equation.     Since  the  orbit  of  the  earth  is  an 
ellipse  the  distance  of  the  sun  undergoes  considerable  variations. 
The  farther  the  sun  is  from  the  earth  the  feebler  are  its  disturbing 
effects,  and  in  particular,  the  power  of  lengthening  the  month 
considered  in  the  preceding  article.     Therefore,  as  the  earth  moves 
from  perihelion  to  aphelion  the  disturbance  which  increases  the 
length  of  the  month  will  become  less  and  less;  that  is,  the  length 
of  the  month  will  become  shorter,  or  the  moon's  angular  motion 
will  be  accelerated.     While  the  earth  is  moving  from  aphelion  to 
perihelion  the  moon's  motion  will,  for  the  opposite  reason,  be 
retarded.     This  is  the  Annual  Equation  amounting  to  a  little 
more  than  11',  and  was  discovered  from  observations  by  Tycho 
Brahe  about  1590. 

194.  The  Secular  Acceleration  of  the  Moon's  Mean  Motion. 
In  the  early  part  of  the  18th  century  Halley  found  from  a  com- 
parison of  ancient  and  modern  eclipses  that  the  mean  motion  of 
the  moon  is  gradually  increasing.     Nearly  100  years  later  (1787) 
Laplace  gave  the  explanation  of  it,  showing  that  it  is  caused  by  the 
gradual  average  decrease  of  the  eccentricity  of  the  earth's  orbit, 
which  has  been  going  on  for  many  thousands  of  years  because  of 
perturbations  by  the  other  planets,  and  which  will  continue  for  a 
long  time  yet  before  it  begins  to  increase. 


194]         SECULAR  ACCELERATION   OF   MOON'S   MEAN   MOTION.  349 

One  effect  of  a  change  in  the  eccentricity  of  the  earth's  orbit  is 
to  change  the  average  disturbing  power  of  the  sun  on  the  orbit  of 
the  moon.  It  will  now  be  shown  that  if  the  eccentricity  decreases, 
the  average  disturbing  power  decreases. 

The  effect  upon  the  moon's  period  is  due  almost  entirely  to  the 
normal  component,  because  it  alone  acts  nearly  along  the  radius 
of  the  orbit,  and  therefore  in  this  discussion  consideration  of  the 
tangential  component  may  be  omitted.  The  average  value  of 
N  in  a  revolution  of  the  moon,  for  R  and  U  constant  and  e  placed 
equal  to  zero,  is  found  from  the  second  equation  of  (18)  to  be 


Average  N  =  -  ^S—ll  -  3  cos  2(nt  -  U)]dt 
(19) 


That  is,  the  normal  component  of  the  disturbing  acceleration  on 
the  average  is  very  nearly  proportional  to  the  radius  of  the  moon's 
orbit  and  the  inverse  third  power  of  the  radius  of  the  earth's  orbit. 
But  if  the  earth's  orbit  is  eccentric,  the  result  for  a  whole  year 
depends  upon  the  eccentricity.  When  the  nature  of  the  depend- 
ence of  the  average  N  upon  the  eccentricity  of  the  earth's  orbit 
has  been  found,  the  effect  of  an  increase  or  decrease  in  this  ec- 
centricity can  be  determined. 

Let  N  represent  the  average  N  for  a  year.     Then  it  follows 
from  (19)  that 

»- 


where  P  is  the  earth's  period  of  revolution.     By  the  law  of  areas 
it  follows  that  hdt  =  R2d6;  hence  equation  (20)  becomes 

-  lWSrr2ndB  1  k*Sr  C2"  (1  +  e'  cos 


r2ndB=        1  k*Sr  C2 
J>     R  =        2  Ph  J> 


2  Ph     >     R  2  Ph     >        O'(l  -  e") 


Pha'(l  -  e'2) ' 

where  a'  and  e'  are  the  major  semi-axis  and  eccentricity  of  the  sun's 
orbit.     But  it  follows  from  the  problem  of  two  bodies  that 

, 9   _/! 

h  =  k  V(l  +  m)o'(l  ~  e'2)  i         P  =       ,          -  . 

fc\l  +  m 


350  THE    VARIATION.  [195 

Therefore  the  expression  for  N  becomes 

—  k2Sr 


2a'3(l  -  «")r 

As  ef  decreases  N  numerically  decreases;  therefore,  as  the  eccen- 
tricity of  the  earth's  orbit  decreases,  the  efficiency  of  the  sun  in 
decreasing  the  attraction  of  the  earth  for  the  moon  gradually 
decreases,  and  the  mean  motion  of  the  moon  increases  corre- 
spondingly. The  changes  are  so  small  that  the  alteration  in  the 
orbit  is  almost  inappreciable,  but  in  the  course  of  centuries  the 
longitude  of  the  moon  is  sensibly  increased.  The  theoretical 
amount  of  the  acceleration  is  about  6"  in  a  century.  The  amount 
derived  from  a  discussion  of  eclipses  varies  from  8"  to  12"  '.  It 
has  been  suggested  that  tidal  retardation,  lengthening  the  day, 
has  caused  the  unexplained  part  of  the  apparent  change,  but  the 
subject  seems  to  be  open  yet  to  some  question. 

The  very  long  periodic  variations  in  the  eccentricity  of  the 
earth's  orbit,  whose  effects  upon  the  motion  of  the  moon  have 
just  been  considered,  are  due  to  the  perturbations  of  the  other 
planets.  Although  their  masses  are  so  small  and  they  are  so 
remote  that  their  direct  perturbations  of  the  moon's  motion  are 
almost  insensible,  yet  they  cause  this  and  other  important  varia- 
tions indirectly  through  their  disturbances  of  the  orbit  of  the 
earth.  This  example  of  indirect  action  illustrates  the  great 
intricacy  of  the  problem  of  the  motions  of  the  bodies  of  the  solar 
system,  and  shows  that  methods  of  the  greatest  refinement  must 
be  employed  in  order  to  derive  satisfactory  numerical  results. 

195.  The  Variation.  There  is  another  important  perturbation 
in  the  motion  of  the  moon  which  does  not  depend  upon  the  eccen- 
tricity of  its  orbit.  It  was  discovered  by  Tycho  Brahe,  from 
observation,  about  1590.  Newton  explained  the  cause  of  it  in  the 
Prindpia  by  a  direct  and  elegant  method  which  elicited  the  praise 
of  Laplace. 

It  can  be  explained  most  readily  by  supposing  that  the  undis- 
turbed motion  of  the  moon  is  in  a  circle.  As  has  been  shown,  the 
normal  component  of  the  sun's  disturbing  acceleration  is  negative 
in  the  intervals  ra8Wim2  and  m^m^m^  with  maximum  values  at 
mi  and  ra6.  Suppose  the  undisturbed  motion  at  mi  is  in  a  circle; 
that  is,  that  the  acceleration  due  to  the  attraction  of  the  earth 
exactly  balances  the  centrifugal  acceleration.  There  is  no  tan- 


195] 


THE   VARIATION. 


351 


gential  component  at  this  point  but  a  large  negative  normal  com- 
ponent. The  result  is  that  the  force  which  tends  toward  E  is 
diminished  and  the  orbit  is  less  curved  at  this  point  than  the 
circle.  Therefore  the  moon  will  recede  to  a  greater  distance 
from  the  earth  in  quadrature  than  in  the  circular  orbit.  At  the 
point  ms  the  tangential  component  is  zero,  the  force  which  tends 
toward  E  is  increased,  and  the  curvature  is  greater  than  in  the 
circle.  The  conditions  vary  continuously  from  those  at  mi  to 


Fig.  56. 

those  at  w3  in  the  interval  m\m$.  The  corresponding  changes  in 
the  remainder  of  the  orbit  are  evident.  The  whole  result  is  that 
the  orbit  is  lengthened  in  the  direction  perpendicular  to  the  line 
from  the  earth  to  the  sun.  If  the  sun  is  assumed  to  be  so  far  dis- 
tant that  its  disturbing  effects  in  the  interval  m3m5W7  are  equal 
to  those  in  the  interval  m7mim3,  the  orbit,  under  proper  initial 
conditions,  is  symmetrical  with  respect  to  E  as  a  center,  and 
closely  resembles  an  ellipse  in  form.  This  change  of  form  of  the 
orbit,  and  the  auxiliary  changes  in  the  rate  at  which  the  radius 
vector  sweeps  over  areas,  give  rise  to  an  inequality  in  longitude 
between  the  mean  position  and  the  true  position  of  the  moon 
which  amounts  at  times  to  about  39'  30",  and  is  called  the  variation. 
The  variation  has  an  interesting  and  important  connection 
with  the  modern  methods  in  the  Lunar  Theory,  which  were 
founded  by  G.  W.  Hill  in  his  celebrated  memoirs  in  the  first  volume 
of  the  American  Journal  of  Mathematics,  and  in  the  Acta  Mathe- 
matica,  vol.  vm.  A  complete  account  of  this  method  is  given  in 
Brown's  Lunar  Theory  in  the  chapter  entitled,  Method  with  Reel- 


352  THE   PARALLACTIC   INEQUALITY.  [196 

angular  Coordinates.  Hill  neglected  the  solar  parallax;  that  is,  he 
assumed  that  the  disturbing  force  is  equal  in  corresponding  points 
in  conjunction  with,  and  opposition  to,  the  sun.  Instead  of 
taking  an  ellipse  as  a  first  approximation,  he  took  as  an  inter- 
mediate orbit  that  variational  orbit  which  is  closed  with  respect  to 
axes  rotating  with  the  mean  angular  velocity  of  the  sun,  with  a 
synodic  period  equal  to  the  synodic  period  of  the  moon.  The 
conception  is  not  only  one  of  great  value,  but  the  analysis  was 
made  by  Hill  with  rare  ingenuity  and  elegance. 

196.  The  Parallactic  Inequality.  Since  the  sun  is  only  a  finite 
distance  from  the  earth,  its  disturbing  effects  will  not  be  exactly 
the  same  in  points  symmetrically  situated  with  respect  to  the  line 
m3w7,  but  will  be  greater  on  the  side  m7mim3.  For  example,  if 
the  expansion  of  p~3  in  (17)  is  carried  one  order  farther  so  as  to 

r2 

include  the  terms  of  the  second  order,  that  is  in  ^,  the  part  of  N 
which  is  independent  of  e  is  found  to  be 


(t*-  E7)] 

(22) 

-  ~  [3  cos  (u  -  U)  +  5  cos  3(t*  -  U)]  ----  }  . 

When  u  —  U  =  0  the  term  of  the  second  order  has  the  same  sign 
as  the  first  one,  and  when  u  —  U  =  ir  it  has  the  opposite  sign. 
The  effect  of  this  term  is  relatively  small  because  r  -5-  R  =  .0025 
nearly.  The  terms  which  are  of  the  second  order  introduce  a 
distortion  in  the  variational  orbit,  which  leads  to  an  inequality 
of  about  2'  7"  in  the  longitude  of  the  moon  compared  to  the 
theoretical  position  in  the  variational  orbit.  Since  it  is  due  to 
the  parallax  of  the  sun  it  has  been  called  the  parallactic  inequality. 
Laplace  remarked  that,  when  it  has  been  determined  with  very 
great  accuracy  from  a  long  series  of  observations,  it  will  furnish  a 
satisfactory  method  of  obtaining  the  distance  of  the  sun.  The 
chief  practical  difficulty  is  that  the  troublesome  problem  of  finding 
the  relative  masses  of  the  earth  and  moon  must  be  solved  before 
the  method  can  be  applied.* 

197.  The  Motion  of  the  Line  of  Apsides.  On  account  of  the 
more  complicated  manner  in  which  the  different  components 
affect  the  motion  of  the  line  of  apsides,  the  perturbations  of  this 

*  See  Brown's  Lunar  Theory,  p.  127. 


197] 


MOTION    OF   THE   LINE    OF   APSIDES. 


353 


element  present  greater  difficulties  than  those  heretofore  con- 
sidered. Suppose  first  that  the  line  of  apsides  coincides  with  the 
line  ESj  and  that  the  perigee  is  at  m\.  The  normal  component 
at  mi  is  negative,  and  therefore  (Table,  Art.  182)  produces  a 
retrogression  of  the  line  of  apsides.  On  the  other  hand,  when 
the  moon  is  at  m&  the  negative  normal  component  causes  the 
line  of  apsides  to  advance.  It  was  shown  in  Art.  180  that  the 
effectiveness  of  a  normal  component  acting  while  the  moon 
describes  a  short  arc  at  apogee  is  to  that  of  an  equal  normal 
component  acting  while  an  equal  arc  is  described  at  perigee  as 
a(l  +  e)  is  to  a(l  —  e).  Moreover,  the  second  equation  of  (18) 
shows  that  the  normal  component  varies  directly  as  the  distance 
of  the  moon  from  the  earth.  Therefore  the  normal  component 
is  greater  at  apogee,  and  is  more  effective  in  proportion  to  its 
magnitude,  than  the  corresponding  acceleration  at  perigee.  The 

Normal  Component. 


m 


normal  component  is  positive,  though  comparatively  small,  in  the 
intervals  m»mtfn±  and  memymg.  These  intervals  are  almost  equally 
divided  by  K  and  L  (Fig.  48)  where  the  effect  of  the  normal  com- 
ponent on  the  line  of  apsides  vanishes.  Therefore  it  follows  from 
the  Table  that  the  total  effect  in  these  intervals  is  very  small. 
Hence  when  the  perigee  is  at  mi  the  result  in  a  whole  revolution  is 
to  rotate  the  line  of  apsides  forward  through  a  considerable  angle. 
Similar  reasoning  leads  to  precisely  the  same  results  when  the 
perigee  is  at  m5. 

When  the  perigee  is  at  mi  the  tangential  component  is  equal  in 
24 


354 


MOTION   OF   THE   LINE   OF   APSIDES. 


[197 


numerical  value  and  opposite  in  sign  on  opposite  sides  of  the  major 
axis.  Hence  it  follows  from  the  Table  that  the  effects  are  in  the 
same  direction  and  equal  in  magnitude  for  points  symmetrically 
situated  on  opposite  sides  of  the  major  axis.  But  the  effects  in 

Tangential  Component. 


the  second  and  third  quadrants  are  opposite  in  sign  to  those  in 
the  first  and  fourth  quadrants;  moreover,  they  are  a  little  greater 
in  the  second  and  third  quadrants  because  then  r  is  greatest  and 
the  tangential  component,  by  (18),  is  proportional  to  r.  Hence 
when  the  perigee  is  at  mi  the  total  effect  of  the  tangential  compo- 
nent in  a  whole  revolution  is  to  rotate  the  apsides  forward.  Now 
pair  this  with  the  case  where  the  perigee  is  at  w5,  a  condition  which 
will  arise  because  of  the  motion  of  the  sun  even  if  the  apsides  were 
stationary.  Under  these  circumstances  the  apsides  are  rotated 
backward,  and  the  rotations  in  the  two  cases  offset  each  other. 

Suppose  now  that  the  line  of  apsides  is  perpendicular  to  the  line 
ES.  It  is  immaterial  in  this  discussion  at  which  end  of  the  line 
the  perigee  is,  but,  to  fix  the  ideas,  it  will  be  taken  at  ra3.  The 
normal  component  is  positive  in  the  interval  ra2ra3W4,  and,  ac- 
cording to  the  Table,  rotates  the  line  of  apsides  forward.  It  is 
also  positive  in  the  interval  m6m7m8  and  there  rotates  the  line  of 
apsides  backward.  In  the  latter  case  the  disturbing  acceleration 
is  greater,  and  more  effective  for  its  magnitude,  so  that  the  whole 
result  is  a  retrogression.  The  intervals  m8Wim2  and  m^n^m^  in 
which  the  normal  components  are  negative,  are  divided  nearly 


198]  SECONDARY   EFFECTS.  355 

equally  by  L  and  K]  hence  it  is  seen  from  the  Table  that  their 
results  almost  exactly  balance  each  other  in  a  whole  revolution. 
Therefore,  when  the  perigee  is  at  m3,  the  result  of  the  normal  com- 
ponent on  the  line  of  apsides  for  a  whole  revolution  is  a  consider- 
able retrogression. 

When  the  perigee  is  at  ms  the  tangential  component  is  positive 
in  the  interval  w3ra5  and  negative  in  m5ra7.  From  the  Table  it  is 
seen  that  a  positive  T  in  the  interval  W3w5w7  causes  the  line  of 
apsides  to  rotate  forward,  and  a  negative,  backward.  Since  the 
sign  of  T  is  opposite  in  the  two  nearly  equal  parts  of  the  interval 
the  whole  result  upon  the  line  of  apsides  is  very  small.  The  result 
is  the  same  in  the  half  revolution  w7raiW3.  Thus  it  is  seen  that 
the  combined  effects  of  the  normal  and  tangential  components  in  a 
whole  revolution  is  to  rotate  the  line  of  apsides  backward  when  it 
is  perpendicular  to  the  line  from  the  earth  to  the  sun. 

It  was  found  that  the  line  of  apsides  rotates  forward  when  it 
coincides  with  the  line  from  the  earth  to  the  sun.  The  next 
question  to  be  answered  is  whether  the  advance  or  the  retro- 
gression is  the  greater.  It  is  noticed  that  the  total  changes  arising 
from  the  action  of  the  tangential  components  are  the  differences 
of  nearly  equal  tendencies,  and  therefore  small.  The  same  may  be 
said  of  the  normal  components  which  act  in  the  vicinity  of  the 
ends  of  the  minor  axis  of  the  ellipse.  Moreover,  in  the  two 
positions  considered  they  act  in  opposite  directions  so  that  their 
whole  result  is  still  smaller.  The  most  important  changes  arise 
from  the  normal  components  which  act  in  the  vicinity  of  the  ends 
of  the  major  axis.  It  follows  from  the  second  equation  of  (18) 
that  in  the  first  case,  in  which  the  line  of  apsides  advances,  they 
are  about  twice  as  great  as  in  the  second,  in  which  the  line  of  apsides 
regresses.  Therefore,  the  whole  change  for  the  two  positions  of 
the  line  of  apsides  is  an  advance.  The  results  for  the  positions 
near  the  two  considered  will  be  similar,  but  less  in  amount  up  to 
some  intermediate  points,  where  the  rotation  of  the  line  of  apsides 
in  a  whole  revolution  of  the  moon  will  be  zero.  From  the  way  in 
which  the  tangential  components  change  sign  (Fig.  58)  it  is  evident 
that  these  points  will  be  nearer  to  m3  and  ra?  than  to  mi  and  ra5; 
therefore  the  average  results  for  all  possible  positions  of  the  perigee 
is  an  advance  in  the  line  of  apsides. 

198.  Secondary  Effects.  The  results  thus  far  have  been  derived 
as  though  the  sun  were  stationary.  It  moves,  however,  in  the 
same  direction  as  the  moon.  It  has  been  shown  that  when  the 


356  PERTURBATIONS   OF   THE   ECCENTRICITY.  [199 

moon  is  near  apogee  and  the  sun  near  the  line  of  apsides,  the 
normal  component  makes  the  apsides  advance.  This  advance 
tends  to  preserve  the  relation  of  the  orbit  with  reference  to  the 
position  of  the  sun,  and  the  advance  of  the  apsides  is  prolonged  and 
increased.  On  the  other  hand,  when  the  moon  is  at  perigee  and 
the  sun  near  the  line  of  apsides  the  line  of  apsides  moves  back- 
ward; the  sun  moving  one  way  and  the  line  of  apsides  the  other, 
this  particular  relation  of  the  sun  and  the  moon's  orbit  is  quickly 
destroyed,  and  the  retrogression  is  less  than  it  would  have  been  if 
the  sun  had  remained  stationary.  In  a  similar  manner,  for  every 
relative  position  of  the  line  of  apsides,  the  advance  is  increased 
and  the  retrogression  is  decreased. 

There  is  another  important  secondary  effect  which  depends 
upon  the  tangential  component  and  is  independent  of  the  motion 
of  the  sun.  As  an  example,  take  the  case  in  which  the  line  of 
apsides  passes  through  the  sun  with  the  perigee  at  m\.  The 
tangential  component  in  w3ra5  is  positive,  and,  according  to  the 
Table,  rotates  the  line  of  apsides  forward  until  the  moon  arrives 
at  apogee.  But,  as  the  line  of  apsides  advances,  the  moon  will 
arrive  at  apogee  later,  and  the  effect  of  this  component  will  be 
increased.  When  the  motion  of  the  sun  is  also  included  this 
secondary  effect  becomes  of  still  greater  importance.  In  this 
manner,  perturbation  exaggerates  perturbation,  and  it  is  clear 
what  astronomers  mean  when  they  say  that  nearly  half  the  motion 
of  the  lunar  perigee  is  due  to  the  square  of  the  disturbing  force, 
or  that  it  is  obtained  in  a  second  approximation. 

The  theoretical  determination  of  the  motion  of  the  moon's  line 
of  apsides  has  been  one  of  the  most  troublesome  problems  of 
Celestial  Mechanics;  the  secondary  effects  Escaped  Newton  when 
he  wrote  the  Principia*  and  were  not  explained  until  Clairaut 
developed  his  Lunar  Theory  in  1749.  The  most  successful  and 
masterful  analysis  of  the  subject  yet  made  is  undoubtedly  that  of 
G.  W.  Hill,  in  the  Acta  Mathematica,  vol.  vin.,  which,  for  the 
terms  treated,  leaves  nothing  to  be  desired.  The  line  of  apsides 
of  the  moon's  orbit  makes  a  complete  reyolutionin  about  9|  years. 

199.  Perturbations  of  the  Eccentricity.  Suppose  the  line  of 
apsides  passes  through  the  sun  and  that  the  perigee  is  at  mi. 

*  In  the  manuscripts  which  Newton  left,  and  which  are  now  known  as  the 
Portsmouth  Collection,  having  been  published  but  recently,  a  correct  explana- 
tion of  the  motion  of  the  line  of  apsides  is  given,  and  nearly  correct  numerical 
results  are  obtained. 


199]  PERTURBATIONS   OF   THE   ECCENTRICITY.  357 

From  the  symmetry  of  the  normal  components  with  respect  to 
the  line  ES  and  the  results  given  in  the  Table,  it  follows  that  the 
increase  and  the  decrease  in  the  eccentricity  in  a  complete  revolu- 
tion due  to  this  component,  are  exactly  equal  under  these  cir- 
cumstances. From  the  way  in  which  the  tangential  component 
changes  sign,  and  from  the  results  given  in  the  Table,  it  follows 
that  the  changes  in  the  eccentricity,  due  to  this  component,  also 
exactly  balance.  Therefore  there  is  no  change  in  the  eccentricity 
in  a  complete  revolution  of  the  moon  under  the  conditions  pos- 
tulated. In  a  similar  manner  the  same  results  are  reached  when 
the  perigee  is  at  ra5. 

Suppose  the  line  of  apsides  has  the  direction  ra3w7.  It  can  be 
shown  as  before  that  neither  the  normal  nor  the  tangential  com- 
ponent makes  any  permanent  change  in  the  eccentricity. 

Now  consider  the  case  in  which  the  line  of  apsides  is  in  some 
intermediate  position;  for  simplicity  suppose  it  is  in  the  line  W2w6 
with  the  perigee  at  m2.  Consider  simultaneously  with  this  case 
that  in  which  the  perigee  is  at  m6.  First  consider  only  the  effects 
of  the  normal  component.  It  follows  from  Fig.  57  and  the  Table 
of  Art.  182  that  when  the  perigee  is  at  m2  and  the  moon  is  in  the 
region  m2m4,  the  normal  component  decreases  the  eccentricity; 
and  when  the  perigee  is  at  ra6,  increases  the  eccentricity.  The  two 
effects  largely  destroy  each  other.  But  it  was  shown  in  Art.  181 
that  a  given  normal  component  is  more  effective  in  changing 
the  eccentricity  when  the  moon  is  near  apogee  than  it  is  when  the 
moon  is  correspondingly  near  perigee.  Besides  this,  since  N  is 
proportional  to  r,  as  follows  from  the  second  equation  of  (18),  the 
normal  component  is  larger  the  greater  the  moon's  distance.  For 
both  of  these  reasons,  while  the  moon  is  in  the  arc  ra2w4  the 
increase  of  the  eccentricity  with  the  perigee  at  w6  is  greater  than 
the  decrease  with  the  perigee  at  w2.  The  two  cases  combined 
give  a  small  second  order  residual  increase  in  the  eccentricity 
which  may  be  represented  by  +  Ai6.  Similarly,  while  the  moon 
is  in  the  region  ra4w6  the  effects  of  the  normal  component  on  the 
eccentricity  with  the  perigee  at  ra2  and  w6  are  respectively  an 
increase  and  a  decrease.  On  paying  heed  to  the  relative  positions 
of  the  apogee,  it  is  seen  that  the  combined  effect  on  the  eccentricity 
is  a  second  order  residual  increase  +  A2e.  By  analogous  dis- 
cussions, the  combined  effects  for  the  moon  in  the  arcs  W6w8  and 
ra8ra2  are  the  positive  second  order  residuals  +  A3e  and  +  A4e. 

The  question  arises  whether  the  second  order  residuals  are  not 


358  PERTURBATIONS   OF   THE   ECCENTRICITY.  [199 

in  some  way  destroyed.  In  order  to  show  that  they  also  vanish 
consider  the  case  in  which  the  line  of  apsides  has  a  symmetrically 
opposite  position  with  respect  to  the  line  ES,  that  is,  the  case  in 
which  the  perigee  is  at  w8  or  w4.  When  the  perigee  is  at  m4  and 
the  moon  in  the  region  ra2ra4  the  eccentricity  is  increased  by  the 
normal  component;  when  the  perigee  is  at  ra8,  the  eccentricity  is 
decreased.  The  decrease  in  the  latter  case  is  greater  than  the 
increase  in  the  former  because  when  the  perigee  is  at  ra8  the 
region  ra2w4  is  near  the  apogee.  Therefore  the  combined  effect 
is  a  second  order  decrease  in  the  eccentricity;  and,  since  the  arc 
W2w4  is  not  only  situated  the  same  relatively  with  respect  to  the 
earth  and  sun  but  also  with  respect  to  the  moon's  orbit  as  when 
the  line  of  apsides  was  the  line  ra2w6,  it  follows  that  the  second 
order  decrease  in  the  eccentricity  is  —  Aie.  It  is  found  similarly 
that  when  the  moon  is  in  the  arcs  w4w6,  mtfns,  and  w8w2  the  sums 
of  the  changes  of  the  eccentricity  when  the  perigee  is  at  w4  and  m% 
are  respectively  —  A2e,  —  A3e,  and  —  A4e.  When  these  second 
order  residuals  are  added  to  those  obtained  when  the  line  of 
apsides  was  the  line  W2ra6  the  result  is  zero.  A  corresponding 
discussion  leads  to  the  same  results  for  any  other  position  of  the 
line  of  apsides,  viz.,  it  can  be  paired  with  another  which  is  sym- 
metrically opposite  with  respect  to  the  line  ES  so  that  when  the 
perigee  is  taken  in  both  directions  on  each  line  the  total  effect  of 
the  normal  component  on  the  eccentricity  is  zero.  Therefore  the 
normal  component  in  the  long  run  makes  no  permanent  change  in  the 
eccentricity  of  the  moon's  orbit;  and  a  somewhat  similar  discussion 
establishes  the  same  result  for  the  tangential  component. 

The  sun  does  not,  however,  stand  still  while  the  moon  makes 
its  revolution,  and  the  conditions  which  have  been  assumed  are 
never  exactly  fulfilled.  Nevertheless,  it  is  useful  to  show  how  the 
different  configurations,  even  though  changing  from  instant  to 
instant,  may  be  paired.  In  a  very  great  number  of  revolutions  the 
supplementary  configurations  will  have  occurred  an  equal  number 
of  times,  and  the  eccentricity  will  have  returned  to  its  original 
value.  The  period  required  for  this  cycle  of  change  depends  in  the 
first  place  upon  the  periods  of  the  sun  and  the  moon;  in  the  second 
place,  upon  the  eccentricity  of  the  sun's  orbit  (the  earth's  orbit) ; 
and  lastly,  upon  the  manner  in  which  the  lines  of  apsides  of  the 
sun's  and  moon's  orbits  rotate. 

The  present  system,  with  abundant  geological  and  biological 
evidence  of  a  very  long  existence  for  the  earth  in  at  least  approxi- 


200]  THE   EVECTION.  359 

mately  its  present  condition,  shows  with  reasonable  certainty  that 
the  system  is  nearly  stable,  if  not  quite.  It  is  an  interesting  fact, 
though,  that  those  two  elements,  the  line  of  nodes  and  the  line  of 
apsides,  which  may  change  continually  in  one  direction  without 
threatening  the  stability  of  the  system  do,  on  the  average,  re- 
spectively retrograde  and  advance  forever. 

200.  The  Evection.  It  has  just  been  shown  that  the  eccentricity 
does  not  change  in  the  long  run;  yet  it  undergoes  periodic  varia- 
tions of  considerable  magnitude  which  give  rise  to  the  largest  lunar 
perturbation,  known  as  the  evection.  At  its  maximum  effect  it 
displaces  the  moon  in  geocentric  longitude  through  an  angle  of 
about  1°  15'  compared  to  its  position  in  the  undisturbed  elliptic 
orbit.  This  variation  was  discovered  by  Hipparchus  and  was 
carefully  observed  by  Ptolemy. 

The  perturbations  of  the  elements,  and  of  the  eccentricity  in 
particular,  depend  upon  two  things,  the  position  of  the  moon  in 
its  orbit,  and  the  position  of  the  moon  with  respect  to  the  earth 
and  sun.  Suppose  the  moon  and  sun  start  in  conjunction  with 
the  perigee  at  mi.  Consider  the  motion  throughout  one  synodic 
revolution.  It  follows  from  the  Table  of  Art.  182  and  Figs.  57 
and  58  that  the  eccentricity  is  not  changing  when  the  moon  is  at 
m\\  that  it  is  decreasing,  or  zero,  when  the  moon  is  at  m2,  Ws, 
and  m4;  that  it  is  not  changing  when  the  moon  is  at  m5;  that  it  is 
increasing,  or  zero,  when  the  moon  is  at  m6,  m7,  and  m8;  and  that 
it  ceases  to  change  when  the  moon  has  returned  to  mt  again. 
This  is  true  only  under  the  hypothesis  that  the  perigee  has  re- 
mained at  mi  throughout  the  whole  revolution;  or,  in  other  words, 
that  the  line  of  apsides  advances  as  fast  as  the  sun  moves  in  its 
orbit.  Now,  the  actual  case  is  that  the  sun  moves  about  8.5 
times  as  fast  as  the  line  of  apsides  rotates.  Since  the  synodic 
period  of  the  moon  is  about  29.5  days  while  the  sun  moves  about 
one  degree  daily,  the  moon  will  be  about  26°  past  its  perigee  when 
it  arrives  at  m\.  What  modification  in  the  conclusions  does  this 
introduce?  The  normal  component  is  negative  and,  in  this  part 
of  the  orbit,  causes  an  increase  in  the  eccentricity,  while  the 
tangential  makes  no  change,  since  it  is  zero.  As  the  moon  pro- 
ceeds past  mi  the  normal  component  becomes  less  in  numerical 
value,  while  the  tangential  component  becomes  negative  and  tends 
to  decrease  the  eccentricity.  The  tendencies  of  the  two  com- 
ponents to  change  the  eccentricity  in  opposite  directions  balance 
when  the  moon  is  at  some  point  between  mi  and  m^  instead  of 


360  SECULAR   VARIATIONS.  [201 

at  Wi,  after  which  the  eccentricity  decreases.  There  is  a  corre- 
sponding advance  of  the  point  ne.ar  ra5  at  which  the  eccentricity 
ceases  to  decrease  and  begins  to  increase.  Similar  conclusions 
are  reached  starting  from  any  other  initial  configuration. 

The  results  may  be  summarized  thus:  The  perturbations  of  the 
sun  decrease  the  eccentricity  of  the  moon's  orbit  somewhat  more 
than  half  of  a  synodical  revolution,  and  then  increase  it  for  an 
equal  time.  These  changes  in  the  eccentricity  cause  deviations 
in  the  geocentric  longitude  from  the  ones  given  by  the  elliptic 
theory,  which  constitute  the  evection.  The  appropriate  methods 
show  that  the  period  of  this  inequality  is  about  31.8  days. 

201.  Gauss'  Method  of  Computing  Secular  Variations.  It  has 
been  shown  in  the  preceding  articles  that  some  of  the  elements, 
such  as  the  line  of  nodes  and  the  line  of  apsides,  vary  in  one 
direction  without  limit.  This  change  is  not  at  a  uniform  rate,  for 
in  addition  to  the  general  variations,  there  are  many  short  period 
oscillations  which  are  of  such  magnitude  that  the  element  fre- 
quently varies  in  the  opposite  direction.  When  the  results  are 
put  into  the  symbols  of  analysis,  the  general  average  advance  is 
represented  by  a  term  proportional  to  the  time,  called  the  secular 
variation,  while  the  deviations  from  this  uniform  change  are 
represented  by  a  sum  of  periodic  terms  having  various  periods  and 
phases.  Thus  it  is  seen  that  the  secular  variations  are  caused  by  a 
sort  of  average  of  the  disturbing  forces  when  the  disturbing  and 
disturbed  bodies  occupy  every  possible  position  with  respect  to 
each  other. 

There  are  other  elements,  such  as  the  inclination  and  the 
eccentricity  which,  though  periodic  in  the  long  run,  vary  con- 
tinuously in  one  direction  on  the  average  for  many  thousands 
,'\  /  of  years.  These  changes  may  be  regarded  as  secular  varia- 
tions also,  and  they  likewise  result  from  a  sort  of  average  of 
perturbations. 

In  1818  Gauss  published  a  memoir  upon  the  theory  of  secular 
variations  based  upon  the  conceptions  just  outlined.  His  method 
has  been  applied  especially  in  the  computation  of  the  secular 
variations  of  the  elements  of  the  planetary  orbits.  Instead  of 
considering  the  motions  of  the  bodies,  Gauss  supposed  that  the 
mass  of  each  planet  is  spread  out  in  an  elliptical  ring  coinciding 
with  its  orbit  in  such  a  manner  that  the  density  at  each  point  is 
inversely  as  the  velocity  with  which  the  body  moves  at  that  point. 
He  then  showed  how  to  compute  the  attraction  of  one  ring  upon 


202]  LONG   PERIOD   INEQUALITIES.  361 

the  other,  and  the  rate  at  which  their  positions  and  shapes  change 
under  the  influence  of  these  forces. 

The  method  of  Gauss  has  been  the  subject  of  quite  a  number  of 
memoirs.  Probably  the  most  useful  for  practical  purposes  is  by 
G.  W.  Hill  in  vol.  i.  of  the  Astronomical  Papers  of  the  American 
Ephemeris  and  Nautical  Almanac.  Hill's  formulas  have  been 
applied  by  Professor  Eric  Doolittle  with  great  success,  the  results 
which  he  obtained  agreeing  very  closely  with  those  found  by 
Leverrier  and  Newcomb  by  entirely  different  methods. 

202.  The  Long  Period  Inequalities.  In  the  theories  of  the 
mutual  perturbations  of  the  planets  very  large  terms  of  long 
periods  occur.  They  arise  only  when  the  periods  of  the  two  bodies 
considered  are  nearly  commejisjjTahle,  and  it  is  easy  to  discover 
their  cause  from  geometrical  considerations. 

Since  the  most  important  variation  occurs  in  the  mutual  per- 
turbations of  Jupiter  and  Saturn  the  explanation  will  be  adapted 
to  that  case.  Five  times  the  period  of  Jupiter  is  a  little  more 
than  twice  the  period  of  Saturn.  Suppose  that  the  two  planets 
are  in  conjunction  at  the  origin  of  time  on  the  line  IQ.  After  five 


Fig.  59. 

revolutions  of  Jupiter  and  two  of  Saturn  they  will  be  in  conjunction 
again  on  a  line  l\  very  near  10,  but  having  a  little  greater  longitude. 
This  continues  indefinitely,  each  conjunction  occurring  at  a  little 
greater  longitude  than  the  preceding.  Conjunctions  occurring 
frequently  at  about  the  same  points  in  the  orbits  cause  very  large 
perturbations,  and  the  Long  Period  is  the  time  which  it  takes  the 
point  of  conjunction  to  make  a  complete  revolution.  In  the  case 


362  PROBLEMS 

of  Jupiter  and  Saturn  it  is  about  918  years.  This  inequality,  which 
is  the  greatest  in  the  longitudes  of  the  planets,  displacing  Jupiter 
21'  and  Saturn  49',  long  baffled  astronomers  in  their  attempts  to 
explain  it  as  a  necessary  consequence  of  the  law  of  gravitation. 
Laplace  finally  made  one  of  his  many  important  contributions  to 
Celestial  Mechanics  by  pointing  out  its  true  cause,  and  showing 
that  theory  and  observation  agree4. 


XXIV.     PROBLEMS. 
1.  Prove  that  the  locus  of  the  point  at  which  the  attractions  of  the  sun 

7?  -\/  Q  Ti1 

and  earth  are  equal  is  a  sphere  whose  radius  is  ~ ^ ,  and  whose  center  is 

o  ~~~  Jbs 

on  the  line  joining  the  sun  and  earth,  at  the  distance  ~ ^  from  the  center 

o  —  -ft 

of  the  earth  opposite  to  the  sun,  where  S  and  E  represent  the  mass  of  the  sun 
and  earth  respectively,  and  R  the  distance  from  the  sun  to  the  earth. 

If  R  =  93,000,000  miles,  and  |  =  330,000,  then 

=  161,550  miles, 
=  281  miles. 


S  -E 

RE 

S-E 


Since  the  moon's  orbit  has  a  radius  of  about  240,000  miles,  it  is  always  at- 
tracted more  by  the  sun  than  by  the  earth. 

2.  The  moon  may  be  regarded  as  revolving  around  the  earth  and  disturbed 
by  the  sun,  or  as  revolving  around  the  sun  and  disturbed  by  the  earth.     As- 
sume that  the  moon's  orbit  is  a  circle,  and  find  the  position  at  which  the 
disturbing  effects  of  the  sun  will  be  a  maximum;  show  that  the  disturbing 
effects  due  to  the  earth,  regarding  the  moon  as  revolving  around  the  sun,  are 
a  minimum  for  the  same  position. 

3.  Find'  the  ratio  of  the  greatest  disturbing  effect  of  the  sun  to  the  least 
disturbing  effect  of  the  earth. 

-4ns.  Let  R  equal  the  distance  from  the  sun  to  the  earth,  p  the  distance 
from  the  sun  to  the  moon,  and  r  the  distance  from  the  earth  to  the  moon; 
then 

^  =  £    rl     &^P*  =  $    ?    ?L±J>  =   0114 
DE       E    p*     ~K>  -r*       E'  p3  '  R  +  r 

4.  Find  the  ratio  of  the  sun's  disturbing  force  at  its  maximum  value  to 
the  attraction  of  the  sun,  and  to  the  attraction  of  the  earth. 


HISTORICAL   SKETCH.  363 

5.  Prove  in  detail  the  conclusion  of  Art.  199  that  the  tangential  com- 
ponent produces  no  secular  changes  in  the  eccentricity  of  the  moon's  orbit. 

6.  Suppose  a  planet  disturbs  the  motion  of  another  planet  which  is  near  to 
the  sun.     Find  the  way  in  which  all  the  elements  of  the  orbit  of  the  inner 
planet  are  changed  for  all  relative  positions  of  the  bodies  in  then*  orbits. 

7.  Show  that,  if  the  rates  of  change  of  the  elements  are  known  when  the 
planet  is  in  a  particular  position  in  its  orbit,  the  intensity  and  direction  of 
the  disturbing  force  can  be  found.      Show  that,  if  it  is  assumed  that  the 
distance  of  the  disturbing  body  from  the  sun  is  known,  its  direction  and  mass 
can  be  found.     (This  is  part  of  the  problem  solved  by  Adams  and  Leverrier 
when  they  predicted  the  apparent  position  of  Neptune  from  the  knowledge  of 
its  perturbations  of  the  motion  of  Uranus.     There  are  troublesome  practical 
difficulties  which  arise  on  account  of  the  minuteness  of  the  quantities  involved 
which  do  not  appear  in  the  simple  statement  given  here.) 


HISTORICAL  SKETCH  AND   BIBLIOGRAPHY. 

The  first  treatment  of  the  Problem  of  Three  Bodies,  as  well  as  of  Two 
Bodies,  was  due  to  Newton.  It  was  given  in  Book  I.,  Section  XL,  of  the 
Principia,  and  it  was  said  by  Airy  to  be  "  the  most  valuable  chapter  that 
was  ever  written  on  physical  science."  It  contained  a  somewhat  complete 
explanation  of  the  variation,  the  parallactic  inequality,  the  annual  equation, 
the  motion  of  the  perigee,  the  perturbations  of  the  eccentricity,  the  revolution 
of  the  nodes,  and  the  perturbations  of  the  inclination.  The  value  of  the  motion 
of  the  lunar  perigee  found  by  Newton  from  theory  was  only  half  that  given 
by  observations.  In  1872,  in  certain  of  Newton's  unpublished  manuscripts, 
known  as  the  Portsmouth  Collection,  it  was  found  that  Newton  had  accounted 
for  the  entire  motion  of  the  perigee  by  including  perturbations  of  the  second 
order.  (See  Art.  198.)  This  work  being  unknown  to  astronomers,  the  motion 
of  the  lunar  perigee  was  not  otherwise  derived  from  theory  until  the  year  1749, 
when  Clairaut  found  the  true  explanation,  after  being  on  the  point  of  sub- 
stituting for  Newton's  law  of  attraction  one  of  the  form  a  =  —2  +  ^-.  Newton 

regarded  the  Lunar  Theory  as  being  very  difficult,  and  he  is  said  to  have  told 
his  friend  Halley  in  despair  that  it  "  made  his  head  ache  and  kept  him  awake 
so  often  that  he  would  think  of  it  no  more." 

Since  the  days  of  Newton  the  methods  of  Analysis  have  succeeded  those 
of  Geometry,  except  in  elementary  explanations  of  the  causes  of  different 
sorts  of  perturbations.  In  the  eighteenth  century  the  development  of  the 
Lunar  Theory,  and  of  Celestial  Mechanics  in  general,  was  almost  entirely  the 
work  of  five  men:  Euler  (1707-1783),  a  Swiss,  born  at  Basle,  living  at  St. 
Petersburg  from  1727  to  1747,  at  Berlin  from  1747  to  1766,  and  at  St.  Peters- 
burg from  1766  to  1783;  Clairaut  (1713-1765),  born  at  Paris,  and  spending 
nearly  all  his  life  in  his  native  city;  D'Alembert  (1717-1783),  also  a  native 
and  an  inhabitant  of  Paris;  Lagrange  (1736-1813),  born  at  Turin,  Italy, 
but  of  French  descent,  Professor  of  Mathematics  in  a  military  school  in  Turin 


364  HISTORICAL   SKETCH. 

from  1753  to  1766,  succeeding  Euler  at  Berlin  and  spending  twenty  years 
there,  going  to  Paris  and  spending  the  remainder  of  his  life  in  the  French 
capital;  and  Laplace  (1749-1827),  son  of  a  French  peasant  of  Beaumont,  in 
Normandy,  Professor  in  1'Ecole  Militaire  and  in  1'Ecole  Normale  in  Paris, 
where  he  spent  most  of  his  life  after  he  was  eighteen  years  of  age.  The  only 
part  of  their  work  which  will  be  mentioned  here  will  be  that  relating  to  the 
Lunar  Theory.  The  account  of  investigations  in  the  general  planetary  theories 
comes  more  properly  in  the  next  chapter. 

There  was  a  general  demand  for  accurate  lunar  tables  in  the  eighteenth 
century  for  the  use  of  navigators  in  determining  their  positions  at  sea.  This, 
together  with  the  fact  that  the  motions  of  the  moon  presented  the  best  test 
of  the  Newtonian  Theory,  induced  the  English  Government  and  a  number  of 
scientific  societies  to  offer  very  substantial  prizes  for  lunar  tables  agreeing 
with  observations  within  certain  narrow  limits.  Euler  published  some  rather 
imperfect  lunar  tables  in  1746.  In  1747,  Clairaut  and  d'Alembert  presented 
to  the  Paris  Academy  on  the  same  day  memoirs  on  the  Lunar  Theory.  Each 
had  trouble  in  explaining  the  motion  of  the  perigee.  As  has  been  stated, 
Clairaut  found  the  source  of  the  difficulty  in  1749,  and  it  was  also  discovered 
by  both  Euler  and  d'Alembert  a  little  later.  Clairaut  won  the  prize  offered 
by  the  St.  Petersburg  Academy  in  1752  for  his  Theorie  de  la  Lune.  Both  he 
and  d'Alembert  published  theories  and  numerical  tables  in  1754.  They  were 
revised  and  extended  later.  Euler  published  a  Lunar  Theory  in  1753,  in  the 
appendix  of  which  the  analytical  method  of  the  variation  of  the  elements  was 
partially  worked  out.  Tobias  Mayer  (1723-1762),  of  Gottingen,  compared 
Euler's  tables  with  observations  and  corrected  them  so  successfully  that  he 
and  Euler  were  each  granted  a  reward  of  £3000  by  the  English  Government. 
In  1772  Euler  published  a  second  Lunar  Theory  which  possessed  many  new 
features  of  great  importance. 

Lagrange  did  little  in  the  Lunar  Theory  except  to  point  out  general  methods. 
On  the  other  hand,  Laplace  gave  much  attention  to  this  subject,  and  made 
one  of  his  important  contributions  to  Celestial  Mechanics  in  1787,  when  he 
explained  the  cause  of  the  secular  acceleration  of  the  moon's  mean  motion. 
He  also  proposed  to  determine  the  distance  of  the  sun  from  the  parallactic 
inequality.  Laplace's  theory  is  contained  in  the  third  volume  of  his  Mecanique 
Celeste. 

Damoiseau  (1768-1846)  carried  out  Laplace's  method  to  a  high  degree  of 
approximation  in  1824-28,  and  the  tables  which  he  constructed  were  used 
quite  generally  until  Hansen's  tables  were  constructed  in  1857.  Plana 
(1781-1869)  published  a  theory  in  1832,  similar  in  most  respects  to  that  of 
Laplace.  An  incomplete  theory  was  worked  out  by  Lubbock  (1803-1865)  in 
1830-4.  A  great  advance  along  new  lines  was  made  by  Hansen  (1795- 
1874)  in  1838,  and  again  in  1862-4.  His  tables  published  in  1857  were  very 
generally  adopted  for  Nautical  Almanacs.  De  Ponte"coulant  (1795-1874) 
published  his  Theorie  Analytique  du  Systeme  du  Monde  in  1846.  The  fourth 
volume  contains  his  Lunar  Theory  worked  out  in  detail.  It  is  in  its  essentials 
similar  to  that  of  Lubbock.  A  new  theory  of  great  mathematical  elegance, 
and  carried  out  to  a  very  high  degree  of  approximation,  was  published  by 
Delaunay  (1816-1872)  in  1860  and  1867. 

A  most  remarkable  new  theory  based  on  new  conceptions,  and  developed 


HISTORICAL  SKETCH.  365 

by  new  mathematical  methods,  was  published  by  G.  W.  Hill  in  1878  in  the 
American  Journal  of  Mathematics.  The  first  fundamental  idea  was  to  take 
the  variational  orbit  as  an  approximate  solution  instead  of  the  ellipse.  Ex- 
pressions for  the  coordinates  of  the  variational  orbit  were  developed  with  rare 
mathematical  skill,  and  are  noteworthy  for  the  rapidity  of  their  convergence. 
A  second  approximation  giving  part  of  the  motion  of  the  perigee  was  published 
in  volume  vm.  of  Acta  Mathematica.  This  memoir  contained  the  first  solution 
of  a  linear  differential  equation  having  periodic  coefficients,  and  introduced 
into  mathematics  the  infinite  determinant.  Hill's  researches  have  been 
extended  to  higher  approximations,  and  completed,  by  a  series  of  papers 
published  by  E.  W.  Brown  in  the  American  Journal  of  Mathematics,  vols. 
xiv.,  xv.,  and  xvn.,  and  in  the  Monthly  Notices  of  the  R.A.S.,  LII.,  LIV.,  and 
LV.  As  it  now  stands  the  work  of  Brown  is  numerically  the  most  perfect 
Lunar  Theory  in  existence,  and  from  this  point  of  view  leaves  little  to  be 
desired.  The  motion  of  the  moon's  nodes  was  found  by  Adams  (1819-1892) 
by  methods  similar  to  those  used  by  Hill  in  determining  the  motion  of  the 
perigee. 

For  the  treatment  of  perturbations  from  geometrical  considerations  con- 
sult the  Principia,  Airy's  (1801-1892)  Gravitation,  and  Sir  John  Herschel's 
(1792-1871)  Outlines  of  Astronomy.  For  the  analytical  treatment,  aside 
from  the  original  memoirs  quoted,  one  cannot  do  better  than  to  consult 
Tisserand's  Mecanique  Celeste,  vol.  in.,  and  Brown's  Lunar  Theory.  Both 
volumes  are  most  excellent  ones  in  both  their  contents  and  clearness  of  expo- 
sition. Brown's  Lunar  Theory  especially  is  complete  in  those  points,  such 
as  the  meaning  of  the  constants  employed,  which  are  apt  to  be  somewhat 
obscure  to  one  just  entering  this  field. 


CHAPTER  X. 

PERTURBATIONS— ANALYTICAL   METHOD. 


203.  Introductory  Remarks.  The  subject  of  the  mutual 
perturbations  of  the  motions  of  the  heavenly  bodies  has  been  one 
to  which  many  of  the  great  mathematicians,  from  Newton's  time 
on,  have  devoted  a  great  deal  of  attention.  It  is  needless  to  say 
that  the  problem  is  very  difficult  and  that  many  methods  of 
attacking  it  have  been  devised.  Since  the  general  solutions  of 
the  problem  have  not  been  obtained  it  has  been  necessary  to  treat 
special  classes  of  perturbations  by  special  methods.  It  has  been 
found  convenient  to  divide  the  cases  which  arise  in  the  solar  system 
into  three  general  classes,  (a)  the  Lunar  Theory  and  satellite 
theories;  (6)  the  mutual  perturbations  of  the  planets;  and  (c)  the 
perturbations  of  comets  by  planets.  The  method  which  will  be 
given  in  this  chapter  is  applicable  to  the  planetary  theories,  and 
it  will  be  shown  in  the  proper  places  why  it  is  not  applicable  to  the 
other  cases.  References  were  given  in  the  last  chapter  to  treatises 
on  the  Lunar  Theory,  especially  to  those  of  Tisserand  and  Brown. 
Some  hints  will  be  given  in  this  chapter  on  the  method  of  com- 
puting the  perturbations  of  comets. 

The  chief  difficulties  which  arise  in  getting  an  understanding  of 
the  theories  of  perturbations  come  from  the  large  number  of 
variables  which  it  is  necessary  to  use,  and  the  very  long  trans- 
formations which  must  be  made,  in  order  to  put  the  equations  in  a 
form  suitable  for  numerical  computations.  It  is  not  possible, 
because  of  the  lack  of  space,  to  develop  here  in  detail  the  explicit 
expressions  adapted  to  computation;  and,  indeed,  it  is  not  desired 
to  emphasize  this  part,  for  it  is  much  more  important  to  get  an 
accurate  understanding  of  the  nature  of  the  problem,  the^m^tlie- 
matical  features  of  the  methods  employed,  the  limitations  which 
arlfnecessary,  the  exact  places  where  approximations  are  intro- 
duced, if  at  all,  and  their  character,  the  origin  of  the  various  sorts 
of  terms,  and  the  foundations  upon  which  the  celebrated  theorems 
regarding  the  stability  of  the  solar  system  rest. 

There  are  two  general  methods  of  considering  perturbations, 
(a)  as  the  variations  of  the  coordinates  of  the  various  bodies, 

366 


204]  ILLUSTRATIVE   EXAMPLE.  367 

and  (6)  as  the  variations  of  the  elements  of  their  orbits.  These 
two  conceptions  were  explained  in  the  beginning  of  the  preceding 
chapter.  Their  analytical  development  was  begun  by  Euler  and 
Clairaut  and  was  carried  to  a  high  degree  of  perfection  by  La- 
grange  and  Laplace.  Yet  there  were  points  at  which  pure  as- 
sumptions were  made,  it  having  become  possible  to  establish 
completely  the  legitimacy  of  the  proceedings,  under  the  proper 
restrictions,  only  during  the  latter  half  of  the  nineteenth  century 
by  the  aid  of  the  work  in  pure  Mathematics  of  Cauchy,  Weier- 
strass,  and  Poincare*. 

204.  Illustrative  Example.  The  mathematical  basis  for  the 
theory  of  perturbations  is  often  obscured  by  the  large  number 
of  variables  and  the  complicated  formulas  which  must  be  used. 
Many  of  the  essential  features  of  the  method  of  computing  per- 
turbations can  be  illustrated  by  simpler  examples  which  are  not 
subject  to  the  complexities  of  many  variables  and  involved 
formulas.  One  will  be  selected  in  which  the  physical  relations 
are  also  simple. 

Consider  the  solution  of 


where  k2,  ju,  *>,  and  I  are  positive  constants.  If  /*  and  v  were  zero 
the  differential,  equation  would  be  that  which  defines  simple 
harmonic  motion.  It  arises  in  many  physical  problems,  such  as 
that  of  the  simple  pendulum,  and  of  all  classes  of  musical  instru- 
ments. In  order  to  make  the  interpretation  definite,  suppose  it 
belongs  to  the  problem  of  the  vibrations  of  a  tuning  fork.  The 
first  term  in  the  right  member  may  be  interpreted  as  being  due 
to  the  resistance  of  the  medium  in  which  the  tuning  fork  vibrates. 
It  is  not  asserted,  of  course,  that  the  resistance  to  the  vibrations 
of  a  tuning  fork  varies  as  the  third  power  of  the  velocity.  An 
odd  power  is  taken  so  that  the  differential  equation  will  have  the 
same  form  whether  the  motion  is  in  the  positive  or  negative  direc- 
tion, and  the  first  power  is  not  taken  because  then  the  differen- 
tial equation  would  be  linear  and  could  be  completely  integrated 
in  finite  terms  without  any  difficulty. 

The  left  member  of  equation  (1)  will  be  considered  as  defining 
the  undisturbed  motion  of  the  tuning  fork.  The  first  term  on  the 
right  introduces  a  perturbation  which  depends  upon  the  velocity 


368  ILLUSTRATIVE   EXAMPLE.  [204 

of  the  tuning  fork;  the  second  term  on  the  right  introduces  a 
perturbation  which  is  independent  of  the  position  and  velocity 
of  the  tuning  fork.  The  first  is  analogous  to  the  mutual  per- 
turbations of  the  planets,  which  depend  upon  their  relative  posi- 
tions; the  second  is  more  of  the  nature  of  the  forces  which  produce 
the  tides,  for  they  are  exterior  to  the  earth.  The  tides  are  defined 
by  equations  analogous  to  (1). 

In  order  to  have  equation  (1)  in  the  form  of  the  equations  which 
arise  in  the  theory  of  perturbations,  let 

(2)  x  =  xi,        -jt  =  xz. 

Then  (1)  becomes 

(3) 

ttX'2       .in  91  7. 

-T-  +  k2Xi  =  —  ju£2  -h  v  cos  It. 
at 

The  corresponding  differential  equations  for  undisturbed  motion 
are 


Equations  (4)  are  easily  integrated,  and  their  general  solution  is 
i  =  +  a  cos  kt  +  )8  sin  kt, 


(5) 

—  ka  sin  kt  +  k(3  cos  kt, 

where  a  and  0  are  the  arbitrary  constants  of  integration.  In  the 
terminology  of  Celestial  Mechanics,  a  and  /3  are  the  elements  of 
the  orbit  of  the  tuning  fork. 

Now  consider  the  problem  of  finding  the  solutions  of  equations 
(3).  Physically  speaking,  the  elements  a  and  /3  must  be  so  varied 
that  the  equations  shall  be  satisfied  for  all  values  of  t.  Mathe- 
matically considered,  equations  (5)  are  relations  between  the 
original  dependent  variables  x\  and  xz,  and  the  new  dependent 
variables  a  and  0  which  make  it  possible  to  transform  the  differ- 
ential equations  (3)  from  one  set  of  variables  to  the  other.  This 
would  be  true  whether  (5)  were  solutions  of  (4)  or  not,  but  since 
(5)  are  solutions  of  (4)  and  (4)  are  a  part  of  (3),  a  number  of  terms 
drop  out  after  the  transformation  has  been  made.  On  regarding 


204]  ILLUSTRATIVE  EXAMPLE.  369 

(5)  as  a  set  of  equations  relating  the  variables  x\  and  #2  to  a  and  j3, 
and  making  direct  substitution  in  (3),  it  is  found  that 

>s    *-j7  +  sin    f-rr  =     , 

—  sin  kt~  +  cos  kt-^  =  Ma  sin  kt— 0  cos  Atf?  -f  7  cos  ft. 
at  at  K 

These  equations  are  linear  in  -,-  and  -j-  and  can  be  solved  for  these 

at          at 

derivatives  because  the  determinant  of  their  coefficients  is  unity. 
The  solution  is 

-77  =  —  fjik2[a  sin  kt  —  0  cos  kt]3  sin  kt  —  j  cos  ft  sin  &Z , 

Ctl  /C 

(7)  ; 

-77  =+  nk2[a  sin  kt  —  (3  cos  fc(|3  cos  Atf  +  T  cos  ft  cos  Atf . 
etc  /c 

The  problem  of  solving  (7)  is  as  difficult  as  that  of  solving  (3) 
because  their  right  members  involve  the  unknown  quantities  a 
and  |8  in  as  complicated  manner  as  x\  and  xz  enter  the  right  mem- 
bers of  (3).  But  suppose  //  and  v  are  very  small;  then,  since  they 
enter  as  factors  in  the  right  members  of  equations  (7),  the  depen- 
dent variables  a  and  /3  change  very  slowly.  Consequently,  for  a 
considerable  time  they  will  be  given  with  sufficient  approximation 
if  equations  (7)  are  integrated  regarding  them  as  constants  in  the 
right  members.  To  assist  in  seeing  this  mathematically  consider 
the  simpler  equation 

(8)  ~  =  /ia(l  +  k  cos  kt). 


The  solution  of  this  equation  is 


where  C  is  the  constant  of  integration.     If  the  right  member  of 
this  equation  is  expanded,  the  expression  for  a  becomes 

(9)     a  =  C  \  I  +  /*(<  +  sin  kt)  +^(t  +  sin  kt)*  +•••]. 

If  fjL  is  very  small  and  if  t  is  not  too  great  the  right  member  of  this 
equation  is  nearly  equal  to  its  first  two  terms.  If  it  were  not  for 
the  term  t  which  is  not  in  the  trigonometric  function  no  limitations 
on  t  would  be  necessary.  But  in  general  such  limitations  are 
necessary;  and  in  most  cases,  though  not  in  the  present  one,  they 
are  necessary  in  order  to  secure  convergence  of  the  series. 
25 


370 


ILLUSTRATIVE   EXAMPLE. 


[204 


It  is  observed  that  the  solution  (9)  is  in  reality  a  power  series  in 
the  parameter  /*,  and  the  coefficients  involve  t.  If  it  is  desired 
equation  (8)  can  be  integrated  directly  as  a  power  series  in  /*. 
The  process  is,  in  fact,  a  general  one  which  can  be  used  in  solving 
(7),  and  equations  (10),  which  follow,  are  the  first  terms  of  the 
solution.  The  conditions  of  validity  of  this  method  of  integration 
are  given  in  Art.  207. 

The  fact  that  when  n  is  very  small  a  and  0  may  be  regarded 
as  constants  in  the  right  members  of  (7)  for  not  too  great  values 
of  t  can  be  seen  from  a  physical  illustration.  Consider  the  per- 
turbation theory.  The  changes  in  the  elements  of  an  orbit  depend 
upon  the  elements  of  the  orbits  of  the  mutually  disturbing  bodies 
and  upon  the  relative  positions  of  the  bodies  in  their  orbits.  It  is 
intuitionally  clear  that  only  a  slight  error  in  the  computation  of 
the  mutual  disturbances  of  two  planets  would  be  committed  if 
constant  elements  were  used  which  differ  a  little,  say  a  degree  in 
the  case  of  angular  elements,  from  the  true  slowly  changing  ones. 

If  equations  (7)  are  integrated  regarding  a  and  /3  as  constants 
in  the  right  members,  it  is  found  that 


(10) 


a  =  a0  - 


(a 


(3a2  +  /32)[cos  2kt  - 


-       sin  2kt  +  —   (a2  - 


sn 


2k(l  +  k) 


2k(l  -  k) 


[cos  (I  +  k)t  -  1] 


[cos  (I  -  k)t  -  1], 


2kt-l] 


—  |T  sin  2kt 

v 
2k(l  +  k) 

v 


-  /32)  sin  4to  J 


2k(l  -  k) 


sin  (I  +  k)t 


sin  (I  —  k)t, 


204]  ILLUSTRATIVE  EXAMPLE.  371 

where  ao  and  /3C  are  the  values  of  a  and  0  respectively  at  t  =  0. 
When  these  values  of  a  and  0  are  substituted  in  (5)  the  values  of 
Xi  and  #2  are  determined  approximately  for  all  values  of  t  which 
are  not  too  remote  from  the  initial  time. 

Consider  equations  (10).  The  right  member  of  each  of  them 
has  a  term  which  contains  t  only  as  a  simple  factor,  while  elsewhere 
t  appears  only  in  the  sine  and  cosine  terms.  The  terms  which 
are  proportional  to  t  seem  to  indicate  that  a  and  /3  increase  or 
decrease  indefinitely  with  the  time;  but  it  must  be  remembered 
that  equations  (10)  are  only  approximate  expressions  for  a  and  j8, 
which  are  useful  only  for  a  limited  time.  It  might  be  that  the 
rigorous  expressions  would  contain  higher  powers  of  t,  and  that 
the  sums  would  have  bounded  values,  just  as 

t3       t6 


is  an  expression  whose  numerical  value  does  not  exceed  unity, 
though  a  consideration  of  the  first  term  alone  would  lead  to  the 
conclusion  that  it  becomes  indefinitely  great  with  t.  On  the 
other  hand  the  presence  of  terms  which  increase  proportionally 
to  the  time  may  indicate  an  actual  indefinite  increase  of  the 
elements  a  and  £.  For  example,  it  was  found  in  the  preceding 
chapter  that  the  line  of  nodes  and  the  apsides  of  the  moon's  orbit 
respectively  regress  and  advance  continually.  The  terms  which 
change  proportionally  to  t  i  re  called  secular  terms. 

The  right  members  of  equations  (10)  also  contain  periodic  terms 

having  the  periods  -r  ,  ^r  ,  ,       ,  ,  and  ,  _  ,  .     These  are  known 

as  periodic  terms.  If  Z  and  k  are  nearly  equal  the  terms  which  in- 
volve sines  or  cosines  of  (l—k)t  have  very  long  periods,  and  are  called 
long  period  terms.  Sometimes  terms  arise  which  are  the  products 
of  t  and  periodic  terms.  These  mixed  terms  are  called  Poisson 
terms  because  they  were  encountered  by  Poisson  in  the  discussion 
of  the  variations  of  tfhe  major  axes  of  the  planetary  orbits.  If  (10) 
are  substituted  in  (5)  the  resulting  expressions  for  x\  and  x2  contain 
Poisson  terms  but  no  secular  terms. 

The  physical  interpretation  of  equations  (10)  is  simple.  The 
elements  a  and  /3  continually  decrease  because  of  the  secular  terms; 
that  is,  the  amplitudes  of  the  oscillations  indicated  in  (5)  con- 
tinually diminish.  This  reduction  is  entirely  due  to  the  resistance 
to  the  motion  as  is  shown  by  the  fact  that  these  terms  contain  the 


372 


EQUATIONS  IN  THE  PROBLEM   OF   THREE   BODIES. 


[205 


coefficient  /*  as  a  factor.  There  are  terms  in  x\  and  x%  of  period 
three  times  and  five  times  the  undisturbed  period  which  are  also 
due  to  the  resistance.  And  the  periodic  disturbing  force  intro- 
duces in  a  and  0  terms  whose  periods  depend  both  on  the  period 
of  the  disturbing  force  and  also  on  the  natural  period  of  the  tuning 
fork.  But  it  is  noticed  that  the  periods  of  the  terms  which  they 
introduce  into  the  expressions  for  x\  and  x2  are  the  period  of  the 
disturbing  force  and  the  natural  period  of  the  tuning  fork. 

205.  Equations  in  the  Problem  of  Three  Bodies.  Consider 
the  motion  of  two  planets,  mi  and  m2,  with  respect  to  the  sun,  S. 
Take  the  center  of  the  sun  as  origin  and  let  the  coordinates  of  mi 
be  (xi,  y\,  Zi),  and  of  m2,  (xz,  yZj  z2).  Let  the  distances  of  mi 
and  m2  from  the  sun  be  r\  and  r2  respectively,  and  let  ri,  2  repre- 
sent the  distance  from  mi  to  m2.  Then  the  differential  equations 
of  motion,  as  derived  in  Art.  148,  are 


(11) 


— 


,  2 


N   Zi  Ofii    2 

mi )  — ;  =  m2  — - — : 
ri3  dzi 


,  1 


\   Z2  (7/1/2    1 

m2)  ~  =  mi — 

r23  6z2 


K2,.  =  fc2 


ri1 


J- 


The  right  members  of  equations  (11)  are  multiplied  by  the 
factors  mi  and  m2  which  are  very  small  compared  to  S',  therefore 
they  will  be  of  slight  importance  in  comparison  with  the  terms 
on  the  left  which  come  from  the  attraction  of  the  sun,  at  least  for 
a  considerable  time.  If  mi  and  m2  are  put  equal  to  zero  in  the 
right  members,  the  first  three  equations  and  the  second  three 


205] 


EQUATIONS  IN  THE   PROBLEM   OF  THREE   BODIES. 


373 


form  two  sets  which  are  independent  of  each  other,  and  the 
problem  for  each  set  of  three  equations  reduces  to  that  of  two 
bodies,  and  can  be  completely  solved. 

It  will  be  advantageous  to  reduce  the  six  equations  (11)  of  the 
second  order  to  twelve  of  the  first  order.     Let 


dx 


then  equations  (11)  become 


(12) 


W-*'  =  o> 

41-*--0' 


•s-""1  = 


%     /  =  (fe. 
#  *      d*  * 


^-  +  /b2(>S  +  m1)^  = 
eft  ri3 


dt 


L  + 


dt 


^  + 


+  ^- 


da:i 
dyi 
TzT 


and  similar  equations  in  which  the  subscript  is  2. 

If  the  motions  of  m\  and  m^  were  not  disturbed  by  each  other 
equations  (12)  would  become 


(13) 


p  +  kz(S  +  mi)  2l  =  0, 


=  0, 


and  an  independent  system  of  similar  equations  in  which  the 
subscript    is    2.     Let    Qx  =  i(*i'2  +  y,'2  +  z/2)  -  fe»(jSf" 
then  equations  (13)  take  the  form 


(14) 


(ft        d^i' 

rf^i  _  dQi 
"dT  ~  dW 


^i 
(ft 


dz, 


This  form  of  the  differential  equations  is  convenient  in  connection 
with  the  problem  of  transforming  equations  so  that  the  elliptic 


374 


TRANSFORMATION   OF  VARIABLES. 


[206 


elements  become  the  dependent  variables  whose  values  in  terms 
of  t  are  required. 

206.  Transformation  of  Variables.  In  order  to  avoid  confusion 
in  the  analysis,  and  to  be  able  to  say  where  and  how  the  approxi- 
mations are  introduced,  the  method  of  the  variation  of  param- 
eters must  be  regarded  in  the  first  instance  as  simply  a  trans- 
formation of  variables,  which  is  perfectly  legitimate  for  all  values 
of  the  time  for  which  the  equations  of  transformation  are  valid. 
From  this  point  of  view  the  whole  process  is  mathematically  simple 
and  lucid,  the  only  trouble  arising  from  the  number  of  variables 
involved  and  the  complicated  relations  among  them. 

In  chapter  v.  it  was  shown  how  to  express  the  coordinates  in 
the  Problem  of  Two  Bodies  in  terms  of  the  elements  and  the 
time.  Let  01,  •••,  «6  represent  the  elements  of  the  orbit  m\, 
and  0i,  •  •  •  ,  06  those  of  m^.  Then  the  equations  for  the  coordinates 
in  the  Problem  of  Two  Bodies  may  be  written 


(15) 


=  0(oi, 


*2    = 


06,  I), 

oe,  0 » 

oe,  t), 

06,  0, 

06,  0, 

06,  0, 


=  8(alt 

=  4>(oi, 


06,  0, 

"6,  0, 

«6,  0, 

06,  0> 

06,  0, 

06,  0- 


A  transformation  of  variables  in  equations  (12)  will  now  be 
made.  Let  it  be  forgotten  for  the  moment  that  equations  (15) 
are  the  solutions  of  the  Problem  of  Two  Bodies,  and  that  the 
ai  and  0»  are  the  elements  of  the  two  orbits;  but  let  (15)  be  con- 
sidered as  being  the  equations  which  transform  equations  (12)  in 
the  old  variables,  Xi,  yi,  zx,  Xi,  yi,  zi',  xz,  y*,  z2,  xj,  2/2',  z2',  into  an 
equivalent  system  in  the  new  variables,  ai,  •••-,  o6,  0i,  •  •  •  ,  0r>. 
The  transformations  are  effected  by  computing  the  derivatives 
occurring  in  (12)  and  making  direct  substitutions.  The  deriva- 
tives of  equations  (15)  with  respect  to  t  are 


(16) 


206] 


TRANSFORMATION   OF  VARIABLES. 


375 


The  direct  substitution  of  (16)  in  (12)  gives 

**1  -.  x  >  + 
dt  ^ 


dt 


(17)      1 


and  similar  equations  in  z2,  •  •  •  ,  z*',  and 


•   These  equations 


are  linear  in  the  derivatives  -^  and  can  be  solved  for  them,  ex- 

pressing them  in  terms  of  «i,  •  •  •  ,  «6,  0i,  •  •  •  ,  &,  and  <,  provided 
the  determinant  of  their  coefficients  is  distinct  from  zero. 

But  if  equations  (15)  are  the  solution  of  the  problem  of  un- 
disturbed elliptic  motion  equations  (17)  are  greatly  simplified, 
for  it  is  seen  from  (13)  that,  when  ai,  •  •  •  ,  0:5  are  constant, 

—1  —  xi   =  0  for  all  values  of   t  .    The  partial  derivative  -rr  , 
at  dt 


when 


are  regarded  as  variables,  is  identical  with  - 


when  they  are  regarded  as  constants.     Therefore  -r-^  —  x\   =  0; 
and  similarly  — ^-  +  k2(S  -f-  mi)  —3  =  0,  and  similar  equations  in 

ut  7*1 

y  and  z.     As  a  consequence  of  these  relations  equations   (17) 
reduce  to 

A  dx\   dai 


(18) 


^  ox\  aoi  _  ^  ox\    aa{  _ 

&d*<  dt"    U>          &(~   ~~~ 


°yi  aaj  _  n 

«  T, V"j 

oai  dt 


dt 


dt 

V^-  — 
^i  dai    dt 

V*  ^5_L  ^»  = 
t^  6«i   ^ 


dx 


and  similar  equations  in  the  &.     These  equations  are  linear  in  the 


376  TRANSFORMATION    OF   VARIABLES.  [206 

derivatives  —  r-*  and  can  be  solved  for  them  unless  the  determinant 
dt 

of  their  coefficients  is  zero.  But  the  determinant  of  the  linear 
system  (18)  is  the  Jacobian  of  the  first  set  of  equations  (15)  with 
respect  to  «i,  •  •  •  ,  «6,  and  cannot  vanish  if  these  functions  are 
independent  and  give  a  simple  and  unique  determination  of  the 
elements.*  These  functions  are  independent,  and  in  general  they 
give  simple  and  unique  values  for  the  elements  since  they  are  the 
expressions  for  the  coordinates  in  the  Problem  of  Two  Bodies. 
The  problem  of  determining  the  elements  from  the  values  of  the 
coordinates  and  components  of  velocity  was  solved  in  chap.  v. 
If  mz  =  0  equations  (18)  are  linear  and  homogeneous,  and  since 

the  determinant  is  not  zero  they  can  be  satisfied  only  by  -rf  =  0, 

(i  =  1,  •  •  •,  6).     That  is,  the  elements  are  constants,  which,  of 
course,  is  nothing  new. 

On  solving  equations  (18),  it  is  found  that 


1,    •  *  •  ,  «6i    |8l>    '  '  '  j  06)    t})          C'    ~   1)    '  '  '  >  6), 

(19)  "C 

-    '  '       •  •  •   a  •  0,    •  •  •    8  •  t)        (i  =  1    •  •  •   6) 


It  will  be  remembered  that  in  determining  the  coordinates  in 
the  Problem  of  Two  Bodies  the  first  step,  viz.,  the  computation  of 
the  mean  anomaly,  involved  the  mean  motion,  defined  by  the 
equation 

nya8feVS  +  m,          0'  =  1,2). 
a? 

Since  the  HJ  involve  the  masses  of  the  planets  the  right  members  of 
(15),  and  consequently  of  (19),  involve  m\  and  w2  implicitly. 

In  order  to  justify  mathematically  the  precise  method  of  inte- 
grating equations  (19)  which  is  employed  by  astronomers,  some 
remarks  are  necessary  upon  m\  and  w2.  In  those  places  where 
they  occur  implicitly  in  the  functions  (pi  and  ^i  they  will  be 
regarded  as  fixed  numbers;  as  they  appear  as  factors  of  the  \f/i 
and  ^i  respectively  they  will  be  regarded  as  parameters  in  powers 
of  which  the  solutions  may  be  expanded.  Such  a  generalization 
of  parameters  is  clearly  permissible  because,  if  a  function  involves 
a  parameter  in  two  different  ways,  there  is  no  reason  why  it  may 

*  See  Baltzer's  Determinanten,  p.  141. 


207]  METHOD   OF  SOLUTION.  377 

not  be  expanded  with  respect  to  the  parameter  so  far  as  it  is 
involved  in  one  way  and  not  with  respect  to  it  as  it  is  involved  in 
the  other.  If  the  function,  instead  of  being  given  explicitly,  is 
denned  by  a  set  of  differential  equations  the  same  things  regarding 
the  expansions  in  terms  of  parameters  are  true.  If  the  attractions 
of  bodies  depended  on  something  besides  their  masses  (measured 
by  their  inertias)  and  their  distances,  as  for  example,  on  their 
rates  of  rotation  or  temperatures,  then  mi  and  m2  so  far  as  they 
enter  in  the  <pi  and  \f/i  implicitly  through  n\  and  nz,  where  they 
would  be  defined  numerically  by  their  individual  mutual  attrac- 
tions for  the  sun,  would  be  different  from  their  values  where  they 
occur  as  factors  of  the  pi  and  i^t-,  for  in  the  latter  places  they 
would  be  defined  by  their  attractions  for  each  other. 

Hence,  the  values  of  the  masses  m\  and  ra2  entering  implicitly 
in  equations  (15)  and  (19)  are  treated  as  fixed  numbers,  given  in 
advance,  and  do  not  need  to  be  retained  explicitly;  on  the  other 
hand,  the  m\  and  w2  which  are  factors  of  the  perturbing  terms  of 
the  equations  are  retained  explicitly,  being  supposed  capable  of 
taking  any  values  not  exceeding  certain  limits. 

207.  Method  of  Solution.  Equations  (11)  are  the  general 
differential  equations  of  motion  for  the  Problem  of  Three  Bodies. 
Equations  (12)  are  equally  general.  No  approximations  were 
introduced  in  making  the  transformation  of  variables  by  (15); 
therefore  equations  (19)  are  general  and  rigorous.  The  difference 
is  that  if  (19)  were  integrated  the  elements  would  be  found  instead 
of  the  coordinates  as  in  (11),  but  as  the  latter  can  always  be 
found  from  the  former  this  must  be  regarded  as  the  solution  of  the 
problem. 

Instead  of  interrupting  the  course  of  mathematical  reasoning  by 
working  out  the  explicit  forms  of  (19),  it  will  be  preferable  to  show 
first  by  what  methods  they  are  solved.  Explicit  mention  will  be 
made  at  the  appropriate  times  of  all  points  at  which  assumptions 
or  approximations  are  made. 

When  mi  and  w2  are  very  small  compared  to  $,  as  they  are  in 
the  solar  system,  the  orbits  are  very  nearly  fixed  ellipses,  and 
therefore  a{  and  /3t-  change  very  slowly.  Consequently  if  they 
were  regarded  as  constants  in  the  right  members  of  (19)  and  the 
equations  integrated,  approximate  values  of  the  <*t  and  the  & 
would  be  obtained  for  values  of  t  not  too  remote  from  the  initial 
time.  This  is  the  method  adopted  in  the  illustrative  example 


378  METHOD    OF   SOLUTION.  [207 

of  the  preceding  article,  and  has  been  the  point  of  view  often 
taken  by  astronomers,  especially  in  the  pioneer  days  of  Celestial 
Mechanics.  But  any  theory  which  is  only  approximate,  even 
though  it  is  numerically  adequate,  does  not  measure  up  to  the 
ideals  of  science. 

Equations  (19)  are  of  the  type  which  Cauchy  and  Poincare*  have 
shown  can  be  integrated  as  power  series  in  mi  and  m2.  Cauchy 
proved  that  m\,  ra2;  and  t  can  all  be  taken  so  small  that  the  series 
converge.  Poincare*  proved  the  more  general  theorem*  that  if 
the  orbits  in  which  the  bodies  are  instantaneously  moving  at  the 
initial  time  do  not  intersect,  then  for  any  finite  range  of  values 
of  t,  the  mi  and  ra2  can  be  taken  so  small  that  the  solutions 
converge  for  every  value  of  t  in  the  interval.  However,  the 
masses  cannot  be  chosen  arbitrarily  small  but  are  given  by 
Nature.  Hence  the  practical  importance  of  the  additional  the- 
orem that,  whatever  the  values  of  mi  and  w2,  there  exists  a  range 
for  t  so  restricted  that  the  solutions  of  equations  (19)  as  power 
series  in  the  parameters  mi  and  mz  converge  for  every  value  of  t  in 
the  range.  In  general,  the  larger  the  values  of  the  parameters 
the  more  restricted  the  range.  This  is,  of  course,  a  special  case  of 
a  general  theorem  respecting  the  expansion  of  solutions  of  differ- 
ential equations  of  the  type  to  which  (19)  belong  as  power  series 
in  parameters.! 

It  follows  from  the  last  theorem  quoted  that,  if  the  range  of  t  is 
not  too  great,  the  solutions  of  equations  (19)  can  be  expressed  in 
convergent  power  series  in  mi  and  w«,  of  the  form 


(20) 

j=0  k=0 

where  the  superfixes  on  the  ai  and  ftt  simply  indicate  the  order  of 
the  coefficient.  The  a^k)  and  fr°''  k)  are  functions  of  the  time 
which  are  to  be  determined.  It  has  been  customary  in  the  theory 
of  perturbations  to  assume  without  proof  that  this  expansion  is 
valid  for  any  desired  length  of  time.  As  has  been  stated,  it  can  be 
proved  that  it  is  valid  for  a  sufficiently  small  interval  of  time; 
but  as  the  method  of  demonstration  gives  only  a  limit  within 
which  the  series  certainly  converge,  and  not  the  longest  time 

*  Les  Methodes  Nouvelles  de  la  Mccanique  Celeste,  vol.  I.,  p.  58. 
fSee  Picard's  Traite  d' Analyse,  vol.  11.,  chap.  XL,  and  vol.  HI. 


207] 


METHOD    OF   SOLUTION. 


379 


during  which  they  converge,  and  as  the  limit  is  almost  certainly  far 
too  small,  it  has  never  been  computed.  It  is  to  be  understood, 
therefore,  that  the  method  which  is  just  to  be  explained,  is  valid 
for  a  certain  interval  of  time,  which  in  the  planetary  theories  is 
doubtless  several  hundreds  of  years. 

On  substituting  (20)  in  (19)  and  developing  with  respect  to 
MI  and  7?i2,  it  is  found  that 


dat-<°'°> 


(21) 


dt 


efi 


dt 


dt 


dt 


;  0 


a,- 


d(f>i 


+  higher  powers  in  m\  and  w2, 

/7fl.(o,i)  /7fl.(i,o) 


dt 


dt 


0)  .    Q    (0 ,  0)  Q    (0 ,  0)  .    A 

;    HI  >  >    Mo  >    «v 


+  higher  powers  in  mi  and  w2,       (i—  1,  •  •  • ,  6). 


In  the  partial  derivatives  it  is  to  be  understood  that  at-  and  jS»  are 
replaced  by  «t(0>  0)  and  jS/0- 0)  respectively.  If  wi  and  m2  were 
not  regarded  as  fixed  numbers  in  the  left  members  of  equations 

•\  i  _       r\  i  _ 

(11),  fa,  \l/i,  -^ ,  r^,  etc.,  would  have  to  be  developed  as  power 

OCX]      Opj 


380 


METHOD    OF   SOLUTION. 


[207 


series  in  Wi  and  ra2,  thus  adding  greatly  to  the  complexity  of  the 
work. 

Within  the  limits  of  convergence  the  coefficients  of  like  powers 
of  mi  and  w2  on  the  two  sides  of  the  equations  are  equal.  Hence, 
on  equating  them,  it  follows  that 


(22) 


(23) 


dt 


(i-1,  ..-,  6), 


>,0)       ...       ,^(0,0).    /Q,  (0,0) 
,  ,    «6  ,    Pi  , 


=  0, 


dt 


(24) 


dt          fa 

^  =  0, 


eft 


On  integrating  equations  (22)  and  substituting  the  values  of 
af(0-0)  and  fr(0-0)  thus  obtained  in  (23),  the  latter  are  reduced  to 
quadratures  and  can  be  integrated;  on  integrating  (23)  and  sub- 


stituting the  expressions  for  a^0-1),  at-(1'0), 


U,  0t(1'0)  in  (24), 


the  latter  are  reduced  to  quadratures  and  can  be  integrated; 
and  this  process  can  be  continued  indefinitely.     In  this  manner 


208] 


CONSTANTS  OF  INTEGRATION. 


381 


the  coefficients  of  the  series  (20)  can  be  determined,  and  the 
values  of  on  and  &i  can  be  found  to  any  desired  degree  of  precision 
for  values  of  the  time  for  which  the  series  converge. 

208.  Determination  of  the  Constants  of  Integration.    A  new 

constant  of  integration  is  introduced  when  equations  (22),  (23),  •  •  • 
are  integrated  for  each  oti(3''k\  Pi(i'k).  These  constants  will  now 
be  determined. 

Let  the  constant  which  is  introduced  with  the  a;0-  k)  be  denoted 
by  -  ai^'V  and  with  the  ft0'-*0,  by  —  &<<'•*>.     Since  the  first  set 
of  differential  equations  have  m2  as  a  factor  in  their  right  members, 
while  the  second  set  have  mi  as  a  factor,  it  follows  that 
«.</,<»  =  0,  (/,<»,         y  =  0,  ••-«>), 
0.(o,*>  =  6.(o,*)>         (fc  =  0,  •••oo). 

Since  the  a;0'-*0  and  j8t°'fc)  are  defined  by  quadratures  all  the 
constants  of  integration  are  simply  added  to  functions  of  t.  That 
is,  the  oti(i'  fc)  and  /3i(/i  fc)  have  the  form 


« 


</,*> 


.  *> 


(0  -  at- 


Therefore  equations  (20)  become 


(25) 


Let  the  values  of  on  and  ft  at  £  =  £0  be 
Then,  at  ^  =  tQ,  equations  (25)  become 


and  /3i(0)  respectively. 


Since  these  equations  must  be  true  for  all  values  of  mi  and  m2 
below  certain  limits,  the  coefficients  of  corresponding  powers  of 
mi  and  m2  in  the  right  and  left  members  are  equal;  whence 


(26)- 


^.(0,0)    = 


a.(/.o>  =0, 

^.(O.t)    =    0, 

=  0, 
=0,         ( 


1,    ...oo), 

1,      .  •  .  00), 


382  TERMS   OF   THE   FIRST  ORDER.  [209 

Since  all  the  terms  of  the  right  members  of  (25)  except  the  first 
vanish  at  t  =  to,  it  follows  that  a;(0'0)  and  /3i(0i0)  are  the  osculating 
elements  [Art.  172]  of  the  orbits  of  mi  and  mz  respectively  at  the 
time  t  =  to,  and  that  the  other  coefficients  of  (20)  are  the  definite 
integrals  of  the  differential  equations  which  define  them  taken 
between  the  limits  t  =  tQ  and  t  =  t. 

209.  The  Terms  of  the  First  Order.  The  terms  of  the  first 
order  with  respect  to  the  masses  are  defined  by  equations  (23). 
Since  the  terms  of  order  zero  are  the  osculating  elements  at  to, 
the  differential  equations  become 

v  (0) .   a  (0)     ,          a  (0) .  f\ 

-*6       7    Pi       j  j  P6       j    ^jj 

(27) 

~dt 

The  right  members  of  these  equations  are  proportional  to  the  rates 
at  which  the  several  elements  of  the  orbits  of  the  two  planets 
would  vary  at  any  time  t,  if  the  two  planets  were  moving  at  that 
instant  strictly  in  the  original  ellipses.  The  integrals  of  (27)  are, 
therefore,  the  sums  of  the  instantaneous  effects;  or,  in  other  words, 
they  are  the  sums  of  the  changes  which  would  be  produced  if  the 
forces  and  their  instantaneous  results  were  always  exactly  equal 
to  those  in  the  undisturbed  orbits.  Of  course  the  perturbations 
modify  these  conditions  and  produce  secondary,  tertiary,  and 
higher  order  effects.  They  are  included  in  the  coefficients  of 
higher  powers  of  mi  and  ra2  in  (20). 

The  quantities  ai(0il)  and  /3i(li0)  are  usually  called  perturbations 
of  the  first  order  with  respect  to  the  masses.  The  reason  is  clearly 
because  they  are  the  coefficients  of  the  first  powers  of  the  masses 
in  the  series  (20).  In  the  planetary  theories  it  is  not  necessary  to 
go  to  perturbations  of  higher  orders  except  in  the  case  of  the 
larger  planets  which  are  near  each  other,  and  then  comparatively 
few  terms  are  great  enough  to  be  sensible.  It  is  not  necessary  in 
the  present  state  of  the  planetary  theories  to  include  terms  of  the 
third  order  except  in  the  mutual  perturbations  of  Jupiter  and  Saturn. 

Instead  of  there  being  but  two  planets  and  the  sun  there  are 
eight  planets  and  the  sun,  so  that  the  actual  theory  is  not  quite 
so  simple  as  that  which  has  been  outlined.  Yet,  as  will  be  shown, 
the  increased  complexity  comes  chiefly  in  the  perturbations  of 
higher  orders.  If  there  were  a  third  planet  w3  whose  orbit  had 
the  elements  71,  •  •  •,  70,  equations  (23)  would  become 


210] 


TERMS  OF  THE  SECOND  ORDER. 


383 


(28) 


=TF      °' 

da((0.1.<» 

~~dT     =<i>< 

/7/v-  (o.o.i) 


ftt 

^.(1,0,0) 

~~dT~ 

^t.  (0,1,0) 
^.(0,0,1)    _ 


.  (o.o.i) 


=  0. 


If  there  were  more  planets  more  equations  of  the  same  type 
would  be  added.  Consider  the  perturbations  of  the  first  order  of 
the  elements  of  the  orbits  m\\  they  are  composed  of  two  distinct 
parts  given  by  the  second  and  third  equations  of  (28),  one  coming 
from  the  attraction  of  ra2,  and  the  other  from  the  attraction  of  m3. 
Therefore,  the  statement  of  astronomers  that  the  perturbing  ef- 
fects  of  the  various  planets  may  be  considered  separately,  is  true 
for  the  perturbations  of  the  first  order  with  respect  to  the  masses. 

210.  The  Terms  of  the  Second  Order.  It  has  been  shown  that 
a.a.o)  =  a.(2,o)  =  ^.(o.i)  =  0.(o.2)  =  o;  therefore  it  follows  from 
(24)  that  the  terms  of  the  second  order  with  respect  to  the  masses 
are  determined  by  the  equations 


(29) 


;  Q 


dt 


, 

dt  f={ 


day 


„  ( 

i      '       J 


dt 


a, 


dt 


0) 
' 


384  TERMS   OF   THE   SECOND   ORDER.  [210 

The  perturbations  of  the  first  order  are  those  which  would  result 
if  the  disturbing  forces  at  every  instant  were  the  same  as  they 
would  be  if  the  bodies  were  moving  in  the  original  ellipses.  If  the 
bodies  mi  and  ra2  move  in  curves  differing  from  the  original  ellipses 
the  rates  at  which  the  elements  change  at  every  instant  are  dif- 
ferent from  the  values  given  by  equations  (27) .  The  perturbations 
of  the  elements  of  the  orbit  of  mi  due  to  the  fact  that  w2  departs 
from  its  original  ellipse  by  perturbations  of  the  first  order  are 
given  by  the  equations  of  the  type  of  the  first  of  (29),  for,  if 
p.d.o)  =  0,  it  follows  that  a/1-1*  =  0  also.  The  perturbations  of 
the  elements  of  the  orbit  of  mi  due  to  the  fact  that  mi  departs  from 
its  original  ellipse  by  perturbations  of  the  first  order  are  given  by 
the  equations  of  the  type  of  the  second  of  (29),  for,  if  a/0- 1}  =  0, 
it  follows  that  «i(0'2)  =  0  also.  The  terms  ft'1-1*  and  /3t-(2'0)  in 
the  elements  of  the  orbit  of  mz  arise  from  similar  causes.  Thus  the 
perturbations  of  the  second  order  correct  the  errors  in  the  terms  of 
the  first  order,  and  those  of  the  third  order  correct  the  errors 
in  the  second,  and  so  on. 

As  has  been  said,  the  solutions  expressed  as  power  series  in  the 
masses  converge  if  the  interval  of  time  is  taken  not  too  great. 
In  a  general  way,  the  smaller  the  masses  of  the  planets  the  longer 
the  time  during  which  the  series  converge.  In  the  Lunar  Theory 
the  sun  plays  the  r61e  of  the  disturbing  planet.  Since  its  mass  is 
very  great  compared  to  that  of  the  central  body,  the  earth,  the 
series  in  powers  of  the  masses  as  given  above  would  converge  for 
only  a  very  short  time,  probably  only  a  few  months  instead  of 
years.  Such  a  Lunar  Theory  would  be  entirely  unsatisfactory. 
On  this  account  the  perturbations  in  the  Lunar  Theory  are  de- 
veloped in  powers  of  the  ratio  of  the  distances  of  the  moon  and  the 
sun  from  the  earth,  and  special  artifices  are  employed  to  avoid 
secular  terms  in  all  the  elements  except  the  nodes  and  perigee. 

If  there  is  a  third  planet  the  perturbations  of  the  second  order 
are  considerably  more  complicated.  Let  the  planets  be  Wi,  w2, 
and  ra3,  and  consider  the  perturbations  of  the  second  order  of  the 
elements  of  the  orbit  of  mi.  From  purely  physical  considerations 
it  is  seen  that  the  following  sorts  of  terms  will  arise:  (a)  terms 
arising  from  the  disturbing  action  of  w2  and  w3,  due  respectively 
to  the  perturbations  of  the  first  order  of  the  elements  of  ra2  and  ms 
by  mi;  (b)  terms  arising  from  the  disturbing  action  of  ra2  and  w3, 
due  to  the  perturbations  of  the  first  order  of  the  elements  of  the 
orbit  of  mi  by  w2  and  w3;  (c)  terms  arising  from  the  disturbing 


210]  TERMS   OF  THE   SECOND   ORDER.  385 

action  of  ra2,  due  to  the  perturbations  of  the  first  order  of  the 
elements  of  the  orbit  of  mi  by  ra3;  (d)  terms  arising  from  the 
disturbing  action  of  w2,  due  to  the  perturbations  of  the  first  order 
of  the  elements  of  the  orbit  of  w2  by  w3;  (e)  terms  arising  from  the 
disturbing  action  of  ra3,  due  to  the  perturbations  of  the  first  order 
of  the  elements  of  the  orbit  of  mi  by  ra2;  and  (/)  terms  arising 
from  the  disturbing  action  of  w3,  due  to  the  perturbations  of  the 
first  order  of  the  elements  of  w3  by  w2. 

Under  the  supposition  that  there  are  three  planets,  the  terms  of 
the  second  order  with  respect  to  the  masses  are  found  from  equa- 
tions (19)  and  (20)  to  be 

Lll^xi.o.o^ 


(30) 


dt         f=i         .  dft 

dT~=SJ  ^y~  -^ 


dt 


dt  U  dctj 

Jo;. (0,1,1)  6     QJ  (       (0)      .  .  .  (0).   0(0)      .  .        0(0).  f) 

-dT  =^i~  -£r  «' 


dft 


.0X0,0,1) 


ddj 

^^y,v^i     ,-",a6     ;  71     ,---,76     ;  t)     (n  t  _n) 
i   /  j  i  /ii 


and  similar  equations  for  -^  and  -JT  • 

at  at 

The  first  two  equations  give  the  perturbations  of  the  class  (a), 
for,  <f>i(a,  |8)  and  #»(«,  7)  are  the  portions  of  the  perturbative 
function  given  by  w2  and  m3  respectively,  while  /3/(1>0'0)  and 
yy d.0,0)  are  fae  perturbations  of  the  first  order  of  the  elements  of 
the  orbits  of  w2  and  ra3  by  mi.  Similarly,  the  third  and  fourth 
equations  give  the  perturbations  of  the  class  (&);  the  first  term 
of  the  fifth  equation,  those  of  class  (c) ;  the  second  term,  of  class 
(d) ;  the  third  term,  of  class  (e) ;  and  the  fourth  term,  of  the  class  (/). 
26 


386  PROBLEMS. 

It  appears  from  this  that  the  terms  of  the  second  order  cannot  be 
computed  separately  for  each  of  the  disturbing  planets. 

The  types  of  terms  which  will  arise  in  the  perturbations  of  the 
third  order  can  be  similarly  predicted  from  physical  considera- 
tions, and  the  predictions  can  be  verified  by  a  detailed  discus- 
sion of  the  equations. 


XXV.     PROBLEMS. 

1.  In  equations  (3)  take  the  term  v  cos  It  to  the  left  member  before  starting 
the  integration,  and  include  it  in  equations  (4).     Carry  out  the  whole  process 
of  integration  with  this  variation  in  the  procedure. 

2.  If  equations  (7)  are  integrated  as  power  series  in  /*  and  v,  what  types  of 
functions  of  t  will  arise  in  the  terms  of  the  second  order? 

3.  Write  the  equations  defining  the  terms  of  order  zero,  one,  and  two  in 
the  masses  when  equations  (11)  are  integrated  as  series  in  mi  and  ra2.     Show 
that  the  terms  of  order  zero  are  the  coordinates  that  m\  and  w2  would  have 
if  they  were  particles  moving  around  the  sun  in  ellipses  defined  by  their 
initial  conditions.     Show  that  the  equations  defining  the  terms  of  the  first 
and  higher  orders  are  linear  and  non-homogeneous,  instead  of  being  reduced 
to  quadratures  as  they  are  after  the  method  of  the  variation  of  parameters 
has  been  used. 

4.  Suppose  there  are  four  planets,  m\,  w2,  w3,  m'4;  write  all  the  terms  of 
the  second  order  with  respect  to  the  masses  according  to  (30)  and  interpret 
each. 

5.  Suppose  there  are  two  planets  m\  and  w2;  write  all  of  the  terms  of  the 
third  order  with  respect  to  the  masses  and  interpret  each. 

6.  Suppose  mi  =  ra2  =  w3  and  that  the  planets  are  arranged  in  the  order 
mi,  m2,  ma  with  respect  to  their  distance  from  the  sun.     Show  that  of  the 
perturbations  defined  by  equations  (30)  the  most  important  are  those  given 
by  the  first  and  third  equations  and  the  second  term  of  the  fifth;  that  the 
perturbations  next  in  importance  are  given  by  the  first,  third,  and  fourth 
terms  of  the  fifth  equation;  and  that  the  least  important  are  given  by  the 
second  and  fourth  equations. 


212] 


LAGRANGE  S   BRACKETS. 


387 


211.  Choice  of  Elements.     In  order  to  exhibit  the  manner  in 
which  the  various  sorts  of  terms  enter  in  the  perturbations  of  the 
first_order,  it  will  be  necessary  to  develop  equations  (19)  explicitly. 
This  was  deferred,  on  account  of  the  length  of  the  transformations 
which  are  necessary,  until  a  general  view  of  the  mathematical 
principles  involved  could  be  given. 

If  terms  of  the  first  order  alone  are  considered  the  functions 
<f>i(a,  j8)  can  be  considered  independently  of  $i(a,  &).  Any  inde- 
pendent functions  of  the  elements  may  be  used  in  place  of  the 
ordinary  elements.  In  fact,  one  of  the  elements  already  employed, 
TT  =  co  +  Q>,  is  the  sum  of  two  geometrically  simpler  elements. 
Now  the  form  of  4>i(a,  j8)  will  depend  upon  the  elements  chosen; 
with  certain  elements  they  are  rather  simple,  and  with  others  very 
complicated.  They  will  be  taken  in  the  first  example  which 
follows  so  that  those  functions  shall  become  as  simple  as  possible. 

212.  Lagrange's  Brackets.     Lagrange  has  made  the  following 
transformation  which  greatly  facilitates  the  computation  of  (19). 


Multiply  (18)  by  - 


da\ 


3T»   5T»   IT   respec- 


tively and  add.     The  result  is 


(31) 


daj  I  dxi  dxi  _  dxi  dxi 
dt  [dai  daz   dai  daz 


dyi 
daz 


dai 


dzidzi'      dzi'dzi} 
dai  daz       dai  daz  J 


das  I  dxi  dxi   _  dxi'  dxi  1 

dt   \  dai  das        dai  day  J 


da?  f  dxi  dxir  _  dxi'  dxi   .          ] 
dt   \  dai  da&        dai   da& 


*  '  *  £t  n 

dai 
Lagrange's  brackets  [at-,  a/]  are  defined  by 


(32) 


[a.  a]  =dx^dxi       dxif  dxi      dyi  dy^      dyi'  dyi 
dai  da}-        dai  da}-       dai  da}-         dai  da/ 


i  daj 


388 


PROPERTIES  OF  LAGRANGE's  BRACKETS. 


[213 


Form  the  equations  corresponding  to  (31)  in  a2,  •  •  •,  a6;  the  result- 
ing system  of  equations  is 

,  dai  dRi,  2 


(33) 


, 


§  r  na; 

K«,-]^- 


These  equations  are  equivalent  to  the  system  (18)  and  will  be  used 
in  place  of  them. 

213.  Properties  of  Lagrange's  Brackets.     It  follows  at  once 
from  the  definitions  of  Lagrange's  brackets  that 

•  ["  I     rv 

[a*,  ay]  =   —  [ay,  aj. 

A  more  important  property  is  that  they  do  not  contain  the  time 
explicitly;  that  is, 

d[a{,  ay] 


(34) 


(35) 


dt 


=  0, 


*~  1,  ...,6;  j  =  1,  •••,6), 


as  will  be  proved  immediately. 

Many  complicated  expressions  will  arise  in  the  following  dis- 
cussion which  are  symmetrical  in  x,  y,  and  z.  In  order  to  abbrevi- 
ate the  writing  let  S,  standing  before  a  function  of  x,  indicate  that 
the  same  functions  of  y  and  z  are  to  be  added.  Thus,  for  example, 

In  starting  from  the  definitions  of  the  brackets  and  omitting  the 
subscripts  of  x,  •  •  • ,  z',  which  will  not  be  of  use  in  what  follows, 
it  is  found  that 

{dtx_  dx^       dx_  _dV_        &x?_  dx_  _dx^_  jPx_  1 
daidt  daj       dai  dajdt       daidt  daj       dai  dajdt  J 

*?L+MJ!?L.\ 

idaj       dt   daidaj  J 
I  dx_    dzx'        dx^_ 


dt 


A.sl  —  —  -^-^\  i  of    ^ 

don"   [dt  daj       dt  da,  J  H      1       dt  da 


^AQ/^^.^^I   AQ/^^-^^I 

.da,-'    [dt  da}-        dt    da,)        Bat     [dt  da>        dt    don]' 


213]  PROPERTIES   OF  LAGRANGE*S  BRACKETS.  389 

The  partial  derivatives  of  the  coordinates  with  respect  to  the  time 
are  the  same  in  disturbed  motion  as  the  total  derivatives  in  un- 
disturbed motion.  Therefore  this  equation  becomes  as  a  conse- 
quence of  (14) 


<Mdx_       d^M\  d_nf<Mdx_       dQds_' 

dx  da,       M  &*/  \  ddj     {  dx  dai       dx'  dai 

d    /  dfl  \         d    /  dti  \  _  d2fl            d212 

dai\daj/        da}-  \dai/  daidaj       dajdai 


dt 


which  proves  the  theorem  that  the  brackets  do  not  contain  t 
explicitly.  This  would  iiardlv  be  anticipated  since  each  of  the 
quantities  which  appears  in  the  brackets^  an  explicit  function  of  t. 

Since  the  brackets  do  not  contain  the  time  explicitly  they  may 
be  computed  for  any  epoch  whatever,  and  in  particular  for  t  =  tQ. 
The  equations  become  very  simple  if  the  coordinates  at  the  time 
t  =  tQ  are  taken  for  the  elements  «i,  •  •  •  ,  a6.  This  is  permissible 
since  the  ordinary  elements  are  defined  by  these  quantities,  and 
conversely.  It  must  not  be  supposed  that  they  are  constants; 
they  are  such  quantities  that  if  the  elements  are  computed  from 
them,  and  then  if  the  coordinates  at  any  time  t  are  commuted  using 
these  elements,  the  correct  results  will  be  obtained.  Since  in 
disturbed  motion  the  elements  vary  with  the  time,  the  values  of 
the  coordinates  at  t  =  t0  also  vary.  Otherwise  considered,  if  the 
osculating  elements  at  t  are  used  and  if  the  coordinates  at  the 
time  t  =  t0  are  computed,  it  will  be  found  in  the  case  of  disturbed 
motion  that  the  coordinates  at  t  =  t0  vary,  and  these  values  of 
the  coordinates  are  the  ones  in  question. 

Let  the  coordinates  at  the  time  t  =  tQ  be  XQ)  •  ••',  z0';  then 

o  dx0'       dxQ'  dx<> 


which  equals  zero  because  XQ  is  independent  of  yo  and  XQ.    Simi- 
larly, 

=  [20',  3</]  =  [so,  2/o]  =  0, 


But 

(37)  [s0,  so']  =  [2/o,  </o']  =  [zo,  zol  =  1. 

Therefore  equations  (33)  become  in  this  case 


TRANSFORMATION   TO    ORDINARY   ELEMENTS. 


[214 


dt  " 

dyo  = 
dt  ~ 

dzQ 


dt 

dy0'  _ 
dt 


dzp1 
dt 


=  -  ra2 


Any  system  of  differential  equations  of  the  form  (38)  is  known 
as  a  canonical  system,  and  they  possess  properties  which  make  them 
particularly  valuable  in  theoretical  investigations.  There  is  a 
theorem  that  any  dynamical  problem  in  which  the  forces  can  be 
represented  as  partial  derivatives  of  a  potential  function  can  be  ex- 
pressed in  this  form;  and  if  it  is  possible  to  put  a  problem  in  the 
canonical  form  it  is  possible  to  do  so  in  infinitely  many  systems  of 
dependent  variables. 

If  equations  (38)  were  solved  they  would  give  the  values  of  the 
coordinates  at  to  which  would  have  to  be  used  to  obtain  the  true 
coordinates  at  the  time  t,  under  the  supposition  that  the  planet 
moved  in  an  undisturbed  ellipse  during  t  —  U.  If  the  variables 
were  the  elliptic  elements  the  solutions  of  the  equations  would 
give  the  elements  which  would  have  to  be  used  to  compute  the 
coordinates  at  the  time  £,  when  they  are  supposed  to  have  been 
constant  during  the  interval  t  —  t0.  Thus,  when  the  elements 
have  been  found  the  remainder  of  the  computation  is  that  of 
undisturbed  motion. 

214.  Transformation  to  the  Ordinary  Elements.  The  elements 
used  in  Astronomy  are  not  the  coordinates  at  t  =  to,  but  &,  i,  a, 
e,  TT,  and  T  (or  e  =  IT  —  nT),  which  were  expressed  in  terms  of  the 
initial  conditions  in  Arts.  86,  87,  and  88.  It  will  be  necessary, 
therefore,  to  transform  equations  (38)  to  the  corresponding  ones 
which  involve  only  the  elements  which  are  actually  in  use  by 
astronomers. 

Let  s  represent  any  one  of  the  elements  & ,  i,  a,  e,  TT,  e.  It  may 
be  expressed  symbolically  in  terms  of  the  initial  conditions  by 

(39)  s  =  f(x<>,  yQ,  z0,  XQ',  yQ',  *</)• 

Hence  it  follows  that 


ds=  „  f  df  dx0        df  dx0' } 
dt       *\  dx0  dt  "*"  dx0'  dt    J  ' 


or,  because  of  (38), 


COMPUTATION   OF  LAGRANGE's  BRACKETS. 
ds  ^  f   df  dRit  2         df  dR\, 


215]  COMPUTATION   OF  LAGRANGE's   BRACKETS.  391 

(40) 

The  partial  derivatives  of  Ri,  2  are  expressed  in  terms  of  the 
partial  derivatives  with  respect  to  the  new  variables  by  the 
equations 


(41) 


i   d-Ri,  2  dw       dRi,  2  de 
dir    dx~Q          de     dxQ ' 


i,  2 


afli,  2  di       dRi,  2  aa       afli,  2  de 

--  ' 


,  2 


de 


a€ 


On  carrying  out  the  complicated  computations  of  -r-  ,   •  •  •  ,       7 

OXQ  OZQ 

by  means  of  the  equations  given  in  Arts.  86,  87,  and  88,  and  ex- 
pressing all  the  partial  derivatives  in  terms  of  the  new  variables, 

«\  p  Af? 

the  partial  derivatives     .  1;  2  ,    •  •  •  .    .  *',2  are  found  in  terms  of 

OXQ  OZo 


the  elements  and 


On  substituting  in  (40)  and  - 


expressing  5*- ,  •  •  • ,  ^77  in  terms  of  the  elements,  -r.  is  found  in 
dXQ  OZQ  at 

terms  of  the  elements  and  the  derivatives  of  the  perturbative 
function,  Rit  2,  with  respect  to  the  elements. 

215.  Method  of  Direct  Computation  of  Lagrange's  Brackets. 

The  transformations  required  in  the  method  of  the  preceding  article 
are  very  laborious,  and  the  direct  computation  of  the  brackets, 
though  considerably  involved,  is  to  be  preferred  from  a  practical 
point  of  view.  All  of  the  computation  in  the  transformations  of  this 
sort  might  be  avoided  by  using  canonical  variables;  but,  in  order  to 
employ  them,  a  lengthy  digression  upon  the  properties  of  canonical 
systems  would  be  necessary,  and  such  a  discussion  is  outside  the 
limits  of  this  work.  Still,  the  labor  may  be  notably  reduced  by 
first  taking  elements  somewhat  different  from  those  defined  in 
chapter  v.,  and  then  transforming  to  those  in  more  ordinary  use. 
The  following  is  based  on  Tisserand's  exposition  of  Lagrange's 

method.* 

*  Tisserand's  Mecanique  Celeste,  vol.  i.,  p.  179. 


392 


COMPUTATION    OF   LAGRANGE  S   BRACKETS. 


[215 


Let  the  ^-plane  be  the  plane  of  the  ecliptic,  &>P  the  projection 
of  the  orbit  upon  the  celestial  sphere,  II  the  projection  of  the  peri- 
helion point,  and  P  the  projection  of  the  position  of  the  planet  at 
the  time  t.  In  place  of  TT  and  e,  adopt  the  new  elements  o>  and  a 
defined  by  the  equations 

(42) 


Fig.  60. 

The  following  equations  are  either  given  in  Art.  98,  or  are  ob 
tained  from  Fig.  60  by  the  fundamental  formulas  of  Trigonometry 


(43) 


n  = 


r  = 


COS  V  = 

sin  v  = 
x  = 

y  = 

z  = 


e  sin  #  =  nt  -f- 
a(l  —  e  cos  E), 


IA 


cos  E  —  e 
1  -  e  cos  E ' 

Vl  -  e2  sin  E 
1  -  e  cos  E   ' 

r{cos  (v  +  co)  cos  ft  —  sin  (v  +  co)  sin  ft  cos i] , 
rjcos  (v  +  w)  sin  &  +  sin  (t;  +  co)  cos  ft  cos i], 
r  sin  (v  -f-  co)  sin  i 


215] 


COMPUTATION  OF  LAGRANGE's  BRACKETS. 


393 


From  these  equations  and  their  derivatives  with  respect  to  the 
time  the  partial  derivatives  of  the  coordinates  with  respect  to  the 
elements  can  be  computed.  The  elements  have  been  chosen 
in  such  a  manner  that  they  are  divided  into  two  groups  having 
distinct  properties;  £,  i,  and  co  define  the  position  of  the  plane  of 
motion  and  the  orientation  of  the  orbit  in  the  plane,  and  a,  e, 
and  ff  define  the  dimensions  and  shape  of  the  orbit  and  the  position 
of  the  planet  in  its  orbit.  Therefore  the  coordinates  in  the  orbit 
can  be  expressed  in  terms  of  the  elements  of  the  second  group 
alone,  and  from  them,  the  coordinates  in  space  can  be  found  by 
means  of  the  first  group  alone. 

Take  a  new  system  of  axes  with  the  origin  at  the  sun,  the  positive 
end  of  the  £-axis  directed  to  the  perihelion  point,  the  rj-axis  90° 
forward  in  the  plane  of  the  orbit,  and  the  £-axis  perpendicular  to 
the  plane  of  the  orbit.  Let  the  direction  cosines  between  the 
x-axis  and  the  £,  r;,  and  f-axes  be  a,  a',  a";  between  the  ?/-axis  and 
the  £,  77,  and  £-axes  be  ftt  0',  #";  and  between  the  z-axis  and  the 
£,  TJ,  and  f-axes  be  7,  7'  ',  7".  Then  it  follows  from  Fig.  60  that 


cos   , 
cos  i, 


(44) 


a 

=  COS  CO  COS  ii   — 

sin  co  sin  66 

ft 

=  cos  co  sin  £  + 

sin  co  cos  £ 

7 

=  sin  co  sin  i, 

a' 

=  —  sin  co  cos  £ 

—  cos  co  sin 

0' 

=  —  sin  co  sin  £ 

+  COS  CO  COS 

7' 

=  cos  co  sin  i, 

a" 

=  sin  £  sin  i, 

ff' 

=  —  cos  £  sin  ^, 

v 

=  cos  i. 

cos   , 
cos  i, 


There  exist  among,  these  nine  direction  cosines,  as  can  easily  be 
verified,  the  relations 


(45) 


a2  +  02  + 


a'2  + 


+ 


aa'  +  08'  +  77'  =  0, 
a' a"  +  ft' ft"  +  7V  =  0, 


a  =  0  -  70,  a  =  /T  -  y,  a  =  y  -  y, 
ft  =  7 '<*"  -  a V,  0'  =  y"a  -  a"7,  ft"  =  7«;  -  «7;, 
7  =  a' ft"  -  VOL",  7'  =  a" ft  -  VOL,  7"  =  aft'  -  fta1 '. 


394 


COMPUTATION  OF  LAGEANGE  S  BRACKETS. 


[215 


It  follows  from  (43)  and  (44)  and  the  definition  of  the  new 
system  of  axes  that 


£  =  r  cos  v  =  a(cos  E  —  e),    ij  =  a  Vl  —  e2  sin  E, 


(46)    < 


dt       I  -  e  cos  E ' 

r- 


,  _    —  na  sin  E  _  —  k  V$  +  mi  sin  E 
1  -  e  cos  E  ~     Va(l  -  e  cos  E)    ' 


,  _  ria  Vl  —  e2  cos  #  _  fc  VS  -f  mi  Vl  —  e2  cos 
1  -  e  cos  #  Va(l  -  e  cos  #) 


y'  = 


where  the  accents  on  x,  y,  z,  £,  rj,  and  f  indicate  first  derivatives 
with  respect  to  t. 

The  partial  derivatives  of  a,  •  •  •  ,  y"  with  respect  to  the  elements 
may  be  computed  once  for  all;  they  are  found  from  (44)  to  be 


(47) 


da 


(48) 


da 

d& 

dp 


9ft" 


ir  =  -  %        ¥-  -  0; 

dco  aco 


da 


r  >    —  n  r    —  fi 

30      "'  a"  "   "' 


(49) 


^  -  0: 

dt  =  +  sin  &>  cos  t, 

^'    =  —  cos  &,  cos  i, 
•j  // 
^-  =  y"  sin  co,     -V  =  T"  cos  co,     -~  =  -  sin  i. 


da         ,,    .  da 

T-T  =  a"  sin  co,      —  =  a' 

01  di 


=  ,3"  sin  co,          -  = 


di 


216] 


COMPUTATION  OF  LAGRANGEJS  BRACKETS. 


395 


There  are  as  many  brackets  to  be  computed  as  there  are  combi- 

6! 
nations  of  the  six  elements  taken  two  at  a  time,  or 


2!  4! 


15. 


Three  of  them  involve  elements  of  only  the  first  group;  nine,  one 
element  of  the  first  group  and  one  of  the  second;  and  three,  ele- 
ments of  only  the  second  group.  Let  K  and  L  represent  any  of 
the  elements  of  the  first  group,  &>,  i,  co;  and  P  and  Q  any  of  the 
elements  of  the  second  group,  a,  e,  a.  Then  the  Lagrangian 
brackets  to  be  computed  are 


/  i      <v    ri        o  f  dx  dx'       dx'  dx  ] 

(a)     [X,I,]  =  S{— _-__},     (3  equations), 

„  f  dx  dx'      dx'  dx  ] 
(6)     [JC,P]-S{_  — -__j,     (9  equations), 

/    \         fD      01  C.   f  SX     dX'  dX'  dX    } 

(c)     [P,  Q]  =  S— _-__,     (3  equations). 


(50)   • 


It  is  found  from  (46)  that 

da    ,      da' 
I  dK~  * 
(51) 


dtf 
dK 

dP 


,da' 


and  similar  equations  in  y  and  z. 

216.  Computation  of  [co,  &],  [&,  i],  [i,  «].  Let  S  indicate  that 
the  sum  of  the  functions,  symmetrical  in  a,  j8,  and  7,  is  to  be 
taken.  Then  the  first  equation  of  (50)  becomes  as  a  consequence 
of  (51) 


But  the  law  of  areas  [Art.  89]  gives 


Therefore 

(52)        [K,  L] 


On  computing  the  right  member  of  this  equation  by  means  of  (47), 


396 


COMPUTATION  OP  LAGRANGE^S  BRACKETS. 


[217 


(48),  and  (49),  and  reducing  by  means  of  (45),  the  brackets  in- 
volving elements  of  only  the  first  group  are  found  to  be 

[co,  &]  =  na2  Vl  -  e2  (-  ap  -  a'p'  +  ap  +  a'0')  =  0, 
[&,  i]  =  na2  Vl  -  e2  {(a$"  -  /3a")  QOS  co 

+  (/3V  -a'  ft")  sin  co  } 
(53)  -}  =  na2  Vl  —  e2  (  —  7'  cos  co  —  7  sin  a>) 

=  —  na2  Vl  —  e2  sin  i, 
[i,    co]  =  -  na2  Vl  -  e2  {  (aV  +  jS'/S"  +  T'T")  cos  co 

+  (a"  a  +  P"$  +  y"y)  sin  co}  =  0. 

217.  Computation  of  [K,   P].    The  second  equations  of  (50) 
become,  as  a  consequence  of  (51), 


'       +  pH   y 

^      ^    T  a^ 

It  follows  from  equations  (45),  (47),  (48),  and  (49)  that 


**  AW   '    P  AW   '    *Y  ^v  —  ^> 
oJ±  oJ\.  oA 


>/  r»   / 


Therefore 


218] 


(54) 


COMPUTATION  OF  LAGRANGE's  BRACKETS. 
,  da  ,  a,  80  ,   ,  d 


397 


a>*«  +  *'  *£  +  y  *x  I  ^!jL=jf 

"  dtf  +  ^  d#+Td#  J       aP 


f  da          ,  8(3  ,  dy  1  d  V? 

+  mi[adK+^dK+JdK\-dP 

Let  P  =  a,  e,  a  in  succession.     Then  it  is  found  that 


(55) 


Vo(l  - 


n 

=  0. 


Let  K  =  co,  ^,  i  in  turn  in  (54),  and  make  use  of  (55);  then  it 
is  found  that 


(56)  S 


r       i       na   r, ^ 

[co,a]  =  yVl^62, 


,  6]  = 


—  na?e 


[co,  a]  =  0, 


na 


Vl  -  e2 ' 

j  «]  =  ~2~  Vl  —  e2  cos  t,  [i,  a]  =  0,          [i,  e]  =  0, 

—  naze 


,  e 


_    ^>2 


COS  I, 


,*]=  0,          [i,  (r]  =  0. 


218.  Computation  of  [a,  e],  [e,  a],  [a,  a].    The  third  equation 
of  (50)  becomes,  as  a  consequence  of  (51), 


/  /2 


,      /-, 

-J  )-' 


>\  r  a*  a''  a?  5"'  4-  ^'  a'  ^'^ 

~  - 


COMPUTATION   OF  LAGRANGE'S   BRACKETS. 


[218 


As  a  consequence  of  equations  (45),  the  right  member  of  this  equa- 
tion reduces  to 


m  =  __L_ 

dPdQ      dQdP^dPdQ      dQdP' 


Since  the  brackets  do  not  contain  the  time  explicitly  t  may  be 
given  any  value  after  the  partial  derivatives  have  been  formed. 
The  partial  derivatives  become  the  simplest  when  t  =  T,  the  time 
of  perihelion  passage.  For  this  value  of  t,  E  =  0,  r  =  a(l  —  e), 
and  it  is  found  from  equations  (46)  that* 


(58)  - 


d£  _  1  dr;  _  n     d£'  _  n     dr/'          n     1  -\-  e 

da"       ~e'     d^~  U'     d^=   °'     d^=     "2Vl"-^ 


-^  _  ^7  _  Q       d£   _  _       "•!    — 


1 


na 


de 


de 


=  0, 


de 


l-ej 


dcr  do- 

Then  equation  (57)  gives 

(59)  [a,  e]  =  0,         [e,  <r]  =  0, 


de       1-6 
—  na 


na 


On  making  use  of  the  fact  that  [«», «,]  =  —  [a/,  aj  and  equations 
(53),  (56),  and  (59),  equations  (33)  become 


(60) 


na 


da 


„   n ^  .     .  di  .  na    n r        .  da 

na2  VI  —  e2  sin  t  -3-  +  -p-  VI  —  ez  cos  1 37 
at       2  at 


na2Vl  —  e2  sinf 


dt 


na2e            .de  dRi, 

,                COS  I -J7  =  ^2       ^^ 

/I  _  e2          di  d^ 

1,  2 


*  It  should  be  remembered  that  a  and  e  enter  explicitly  and  also  implicitly 
through  E  and  n,  for  #  is  denned  by  the  equation 


E  -  e  sin  E  =  n(t  -  T)  = 


•\  £  «   |Tt 

Then,  e.  g.,  ~  =  cos  E  —  e  —  a  sin  E  —  =  1  -  e  when  £  =  T7,  etc, 


218] 


COMPUTATION  OF  LAGRANGE*S  BRACKETS. 


399 


(60) 


na   r- dco     na   r. .  d&>      na  da 

~TA         e'dt~~2'*1'  l~dt"~2dt 

na2e     do>      ntfe  cos  i  d&  _         dRi,  2 
^#~di^'  ^T^~dt~'~m2~de~' 


da    ' 


2  (ft  -  da 

These  equations  are  easily  solved  for  the  derivatives,  and  give 


(61) 


na2  Vl  -  e2  sin  i 


^2 


,  2 


di  m2  cos  i        dRit  2  _ 

dt         na2  Vl  —  e2  sin  i    do>         na?  Vl  —  e2  sin  ^    ^  ^ 


nae 


de 


rwi      da 


The  perturbative  function  Rit  2  involves  the  element  a  explicitly, 
and  also  implicitly  through  n  which  enters  only  in  the  combi- 
nation nt  +  cr.  Consequently  the  last  equation  of  (61)  becomes 


da 


m2(l  - 


*\  7")  O/i/i/i         /   m  I?  \ 

o/ti^  _  ^niz  i  Q/II,  2  \  _ 
de          na  \    da    ) 


6//l2   C'-tt'l,  2  CF/t 

?ia      dn    da ' 


where  the  partial  derivative  in  parenthesis  indicates  the  derivative 
is  taken  only  so  far  as  the  parameter  appears  explicitly.      ^ 
It  follows  from  the  combination  nt  +  a  that 


(63) 


2m 


dn 


na 


da 


,  2  _    da 

" 


It  will  be  shown  [Arts.  225-227]  that 


CCT 


*'. 

is  a  sum  of  periodic 


terms;  therefore  cr,  as  denned  by  (62),  contains  terms  which  are 
the  products  of  t  and  trigonometric  terms.  It  is  obvious  that  such 
an  element  is  inconvenient  when  large  values  of  t  are  to  be  used. 


400 


COMPUTATION  OF  LAGRANGE's  BRACKETS. 


[219 


In  order  to  avoid  this  difficulty  Leverrier  used*  in  place  of  a  the 
mean  longitude  from  the  perihelion  as  an  element.     It  is  defined  by 

(64)  I  =  fndt  +  cr, 
whence 

(65)  £-.•  +  £+*. 


Since  n 


it  follows  that 


(66) 


dn  _    _3n  dn  _       3n  da 

~da  ~    ~2  a'  ~dt  ~    ~  2adi' 

Therefore  equation  (65)  becomes,  on  making  use  of  (62), 


-r.  =  n  — 


na?e 


o/ti,  2       2wi2  /  dR\,  2  \ 
de          na  \    da    )  ' 


Since 


or) 

*' 


Off 


,  the  fourth  and  fifth  equations,  where  alone 


the  partial  derivative  of  Rit  2  with  respect  to  a  occurs,  will  not  be 
changed  in  form.  Hence,  if  Us  used  in  place  of  cr  throughout  (61), 
the  equations  will  be  unchanged  in  form,  and  the  partial  deriva- 
tive of  Ri,  2  with  respect  to  a  is  to  be  taken  only  so  far  as  a  occurs 
explicitly. 

219.  Change  from  ft,  co,  and  cr  to  ft,  TT,  and  €.  The  trans- 
formation from  the  elements  ft,  co,  and  a  to  ft,  TT,  and  e  is 
readily  made  because  the  relations  between  the  co  and  cr  and  the 
TT  and  e  are  very  simple.  It  follows  from  the  definitions  of  Arts 
214  and  215  that 

<  V* 


V 


(68) 


whence 


(69) 


e  -  TT; 


dt 


dt  ' 


da)  _  dir 

~dt  ~  ~di  "  ~dt  '. 

da  _  de      dir 
dt  ~  ~dt  ~  ~dt' 


On  solving  (68)  for  & ,  TT,  and  e  in  terms  of  & ,  w,  and  cr,  it  is  found 
that 

*  Annales  de  VObservatoire  de  Paris,  vol.  i.,  p.  255. 


219]  COMPUTATION   OF  LAGRANGE's   BRACKETS.  401 

'    ft=    ft, 

(70)  -        7T    =    CO   +    ft, 

Hence  the  transformations  in  the  partial  derivatives  are  given  by 
the  equations 

afli, 2  _  /a#ll2\  aft 


(71) 


aft 


aft     aft 


lt  2  \_gjr_        /afli.a\  _de 

Tr   /  aft  "*"  \   ae    )  e& 


=  \  aft 
-jjj  =  (  — J^J  j  — °°-  _|_  (  — k*  |  JL  _|_  [  — LJ  j  _i 

aco  \    aft     /   dco         \     dw     /  aco        \     de     /  d<*3 


— li.2  =  /  1?-2  )  — ^i  _|_  /  !»_?  j  _J[  _|_  /  _ — k2  J  _1 

dff  \    aft     /    dff         \     dir     /  do"        \     de      /  dff 

-(^): 

On  substituting  (69)  and  (71)  in  (61)  and  omitting  the  parentheses 
around  the  partial  derivatives,  and  on  solving  for  the  derivatives 
of  the  elements  with  respect  to  t,  it  is  found  that 

dft  ra2  a#i,  2 


(72)  - 


na2  Vl  —  e2  sin  i 


tan  I 


dir 
dt 

da 

dt 

de 
^- 
at 


na2  Vl  —  e2  sin  i  &  ft        na2  Vl  —  • 

i 

L,  2  .  m2  Vl  -  e2  a^i,  2 


it2  i 


H. 


ll  -  e2     di 
dR\,  2 


^  + 


ae 


a€ 


—  02 


-  Vl  - 


,  2 


l  - 


2 


de          m2tan2 
d^ 


na2e         dir 


na^l  — 


nae 


de 


na      da 


27 


402  RECTANGULAR   COMPONENTS   OF   ACCELERATION.  [220 

These  equations,*  together  with  the  corresponding  ones  for  the 
elements  of  the  planet  w2,  constitute  a  rigorous  system  of  differ- 
ential equations  for  the  determination  of  the  motion  of  the  planets 
mi  and  w2  with  respect  to  the  sun  when  there  are  no  other  forces 
than  the  mutual  attractions  of  the  three  bodies. 

If  Rit  2  is  expressed  in  terms  of  the  time  and  the  osculating 
elements  at  the  epoch  to,  equations  (72)  become  the  explicit 
expressions  for  the  tirst  half  of  the  system  (27),  and  define  the 
perturbations  of  the  elements  which  are  of  the  first  order  with 
respect  to  the  masses. 

220.  Introduction  of  Rectangular  Components  of  the  Disturbing 
Acceleration.  Equations  (72)  require  for  their  application  that 
Rit  2  shall  be  expressed  first  in  terms  of  the  elements,  after  which 
the  partial  derivatives  must  be  formed.  In  some  cases,  especially 
in  the  orbits  of  comets,  it  is  advantageous  to  have  the  rates  of 
variation  of  the  elements  expressed  in  terms  of  three  rectangular 
components  of  the  disturbing  acceleration. 

The  disturbing  acceleration  will  be  resolved  into  three  rect- 
angular components  W,  S,  R,  where  W  is  the  component  of 
acceleration  perpendicular  to  the  plane  of  the  orbit  with  the 
positive  direction  toward  the  north  pole;  S  is  the  component  in 
the  plane  of  the  orbit  which  acts  at  right  angles  to  the  radius 
vector  with  the  positive  direction  making  an  angle  less  than  90° 
with  the  direction  of  motion;  R  is  the  component  acting  along  the 
radius  vector  with  the  positive  direction  away  from  the  sun. 
The  components  used  in  the  preceding  chapter  evidently  might  be 
employed  here  instead  of  these,  but  the  resulting  equations  would 
be  less  simple. 

In  order  to  obtain  the  desired  equations  it  is  only  necessary  to 
express  the  partial  derivatives  of  JKi,  2  with  respect  to  the  ele- 
ments in  terms  of  W,  S,  and  R,  and  to  substitute  them  in  (61) 
or  (72),  depending  upon  the  set  of  elements  used.  The  trans- 
formation will  be  made  for  the  elements  used  in  equations  (61). 

The   quantities  m2-^-2,  w2-^,  m2^j-^    are  the    com- 
ox  oy  oz 

ponents  of  the  disturbing  acceleration  parallel  to  the  fixed  axes  of 
reference.     It   follows   from   the   elementary   properties   of   the 

*  The  subscript  1,  which  was  omitted  from  the  coordinates  and  elements  in 
Art.  213,  should  be  replaced  when  the  equations  for  more  than  one  planet  are 
written. 


220] 


RECTANGULAR  COMPONENTS  OF  ACCELERATION. 


403 


resolution  and  composition  of  accelerations  that 


2 


•  i 

is  equal 


to  the  sum  of  the  projections  of  W  ,  S,  and  R  upon  the  x-axis,  and 
similarly  for  the  others. 

Let  u  represent  the  argument  of  the  latitude,  or  the  distance 
from  the  ascending  node  to  the  planet  P,  Fig.  61.     Then  it  follows 


Fig.  61. 

from  the  fundamental  formulas  of  Trigonometry  that 

=  +  R(cos  u  cos  ft  —  sin  u  sin  ft  cos  i) 
—  *S(sin  u  cos  ft  +  cos  u  sin  ft  cos  i) 
+  W  sin  ft  sin  i, 


(73) 


a/L-i,2 

dx 


ra2 


/•)  7? 

,1>2  =  +  R(cos  u  sin  ft  +  sin  u  cos  ft  cos  i) 


, 
dy 


—  S(sm  u  sin  ft  —  cos  u  cos  ft  cos  i) 

—  W  cos  ft  sin  i, 

A  7? 

ra2  — —  =  +  R  sin  u  sin  i  +  S  cos  u  sin  i  -\-  W  cos  i. 
Let  s  represent  any  of  the  elements  ft,  •••,*;  then 


(74) 


L,  2 


dx        S 

as       a?/    as 


, 


__.     .        .      a/ti  2  a /LI  2  a/ti,  2        •       •    /wr»\      i    i. 

The  derivatives     .  '    ,        '    ,     ,      are  given  in  (73)  and  when 
ox         dy          dz 

— ,  — ,  and  —  have  been  found,  the  transformation  can  be  com- 
as  as          as 

pleted  at  once. 


404  RECTANGULAR  COMPONENTS  OF  ACCELERATION. 

It  follows  from  equations  (51)  that 

da' 


[220 


(75) 


dx_ 
dK 

dy_ 
dK 


da 


dP  dp' 

n  TT"     I      */   r\  7^  ) 


dz          dy 
dK      ^dK 


dy' 


!/•</  *-^S          I  f     ^    I 

JP  =  a~dP~*  *  'dP) 

dy_  _  Rd^_,    R,  dr) 
dP~  PdP^P  dP' 

dz  _        d£          ,  df] 

dP~  TaP  +  T  dP> 


where  K  is  any  of  the  elements  &,  i,  «,  and  P  any  of  the  ele- 
ments a,  e,  a.  The  quantities  a,  •  •  •,  y'  are  defined  in  (44),  and 
their  derivatives  are  given  in  (47),  (48),  and  (49);  the  derivatives 

f\t,  *\ 

—ft  and  -r^  are  to  be  computed  from  (46). 

or  or 

It  is  found  after  some  rather  long  but  simple  reductions  that 

mz    .,*'2  =  Sr  cos  i  —  Wr  cos  u  sin  i, 
oii 

mz       ?'2  =  Wr  sin  u, 


(76) 


da  a 

dfli.  2  = 


-Ra  cos  v  +  *S     1  +  -    a  sin  v, 


/i,  2  _      /cae      .        i    c^_ 

o"         "Y^  —  g2  r 


Therefore  equations  (61)  become 
•^  r  sin  u 


(77) 


7—W, 


r  cos 


at      no?  Vl  -  e2 

<fo  _  -  Vl  -  e2  cos 
d<      


i^  +  JZfft^llffln^ 

nae  P  .1 


r  sin  tt  cot  t  ~- 

na2  Vl  -  e2     ' 


(77) 


da 
dt 


2e  sin  v 


de_  _  Vl  —  e2  sin  v 
dt  na 


dcr  1   [2r      1-e2          1  D 

= CQSZ;        # 

d£  na  |_  a          e 


405 


(!_-*, 

nae 


—    1  +  -    sm 


XXVI.     PROBLEMS. 

1.  Find  the  components  S  and  J?  of  this  chapter  in  terms  of  T  and  N, 
which  were  used  in  chapter  ix.,  Art.  174. 


(1  -)-  e  cos  v) 


A.ns. 


e  sin  v 


T  -• 


+  e2  +  2e  cos  v 
1  +  e  cos  v 


+  e2  +  2e  cosv  Vl  +  e2  +  2e  cos  w 


N.  J 


2.  By  means  of  the  equations  of  problem  1  express  the  variations  of  the 
elements  ft,  •  •  •,  a  in  terms  of  T  and  A",  and  verify  all  the  results  contained  in 
the  Table  of  Art.  182. 

3.  Explain  why  -^  contains  a  term  depending  upon  W. 

4.  Suppose  the  disturbed  body  moves  in  a  resisting  medium;  find  the 
equations  for  the  variations  of  the  elements. 


Ans. 


<fc 
Jt 

dt 

=  0, 
=  0, 

dot 

2  V  1  —  e2              sin  y               ^ 

dt 

nae 

V  1  +  e2  +  2e  cos  v 

da 

2Vl  +< 

?  +  2e  cos  v  T 

dt 

n>/ 

l-e2 

de 
dt 

2Vl  -62  (cosy  +  e)m 

naVTT 

e2  +  2e  cos  y 

do- 

2(1 

—  e2)(l  +  e2  +  e  cos  y)  sin  y 

dt 

nae(l 

+  e  cos  y)  V  1  -f  e2  +  2e  cos  y 

5.  Discuss  the  way  in  which  the  elements  vary  in  the  last  problem,  including 
the  values  of  v  for  which  the  maxima  and  minima  in  their  rates  of  change 
occur,  when  T  is  a  constant,  and  when  it  varies  as  the  square  of  the  velocity. 


406  DEVELOPMENT   OF   PERTURBATIVE  FUNCTION.  [221 

6.  Derive  the  equations  corresponding  to  (77)  for  the  elements  &,  i,  TT, 
a,  e,  and  e. 

r  sin  u 


dt 
Ans. 


dt       na  V 1  —  e2  sin  i 
di          r  cos  u 


j^ 
1       e  cos  v 


dt 

221.  Development  of  the  Perturbative  Function.  In  order 
to  apply  equations  (72)  the  perturbative  function  /2lf  2  must  be 
developed  explicitly  in  terms  of  the  elements  and  the  time.  From 
this  point  on  only  perturbations  of  the  first  order  will  be  con- 
sidered; therefore,  in  accordance  with  the  results  of  Art.  208, 
the  elements  which  appear  in  R  i,  2  are  the  osculating  elements  at 
the  time  t0. 

In  the  notation  of  Art.  205  the  perturbative  function  is 

fit, 

(78) 


+  (2/2  -  2/i)2  +  fe  -  zi) 


The  perturbing  forces  evidently  depend  upon  the  mutual 
inclinations  of  the  orbits,  rather  than  upon  their  inclinations 
independently  to  the  fixed  plane  of  reference.  It  will  be  con- 
venient, therefore,  to  develop  Rit  2  in  terms  of  the  mutual  inclina- 
tion. Since  this  angle  is  expressible  in  terms  of  ii,  iz,  &>i,  and  &2, 
the  partial  derivatives  of  Rit  2  with  respect  to  these  elements  will 
depend  in  part  on  their  occurring  implicitly  in  this  angle. 

The  development  of  the  perturbative  function  consists  of  three 
steps:* 

*  There  are  many  more  or  less  important  variations  of  the  method  outlined 
here,  which  is  based  on  the  work  of  Leverrier  in  the  Annales  de  VObservatoire 
de  Paris,  vol  i. 


222] 


DEVELOPMENT  IN  THE  MUTUAL   INCLINATION. 


407 


(a)  Development  of  RI,  2  as  a  power  series  in  the  square  of  the 
sine  of  half  the  mutual  inclination  of  the  orbits. 

(6)  Development  of  the  coefficients  of  the  series  obtained  in 
(a)  into  power  series  in  e\  and  e2. 

(c)  Development  of  the  coefficients  of  the  preceding  series  into 
Fourier  series  in  the  mean  longitudes  of  the  two  planets  and  the 
angular  variables  in,  7r2,  &  i,  and  &  2. 

In  the  little  space  available  here  it  will  not  be  possible  to  give 
more  than  a  general  outline  of  the  operations  which  are  necessary 
to  effect  the  complete  development.  A  detailed  discussion  is 
given  in  Tisserand's  Mecanique  Celeste,  vol.  I.,  chapters  xn.  to 
xvin.  inclusive. 


222.   (a)  Development    of    Ri,z   in   the 
Let  S  represent  the  angle  between  the  radii 


Mutual    Inclination. 

and  r2;  then 


(79) 


+r22  -   2rir2  cos  S)~ 


Fig.  62. 

Let  the  angles  between  r\  and  the  x,  y,  and  z-axes  be  «i,  0i,  71 
respectively,  and  in  the  case  of  r2,  «2,  j82,  and  72.     Then  it  follows 

that 

(80)     Xi  =  ri  cos  «i,        2/1  =  rv  cos  0i,        Zi  =  n  cos  71,    etc., 
and 

£i£2  +  2/i2/2  +  ZiZ2  =  rir2(cos  «i  cos  «2  +  cos  /3i  cos  /32 

+  cos  71  cos  72)  =  r\r<i  cos  S. 


Let  7  represent  the  angle  between  the  two  orbits,  and  T\  and  r2 


408 


DEVELOPMENT  IN   THE   MUTUAL   INCLINATION. 


[222 


the  distances  from  their  ascending  nodes  to  their  point  of  inter- 
section. From  the  spherical  triangle  PiP2C  the  value  of  cos  S  is 
found  to  be 

cos  S  =  cos  (u\  —  TI)  cos  (uz  —  T2) 

+  sin  (ui  —  TI)  sin  (w2  —  T2)  cos  7,     or 
cos  S  =  cos  (ui  —  Uz  +  T2  —  TI) 


(82) 


—  2  sin  (HI  —  TI)  sin  (uz  —  T2)  sin2     > 


—    T2 


7T2   — 


—    T2. 


The  quantities  7,  TI,  and  T2  are  determined  by  the  formulas  of 
Gauss  applied  to  the  triangle  &  i  &>  2C : 


sin  7  sin  TI 
sin  I  sin  T2 


sn 
sin 


sn 
sin 


sin  /  cos  TI  =  sin  ii  cos  t'2  —  cos  ii  sin  iz  cos  (£h 
sin  /  cos  T2  =  —  cos  i\  sin  i2  +  sin  ii  cos  t2  cos  (  &  i  —  ^  2), 
cos  7  =  cos  i'i  cos  iz  +  sin  ii  sin  t'2  cos  (^i 


(83) 


For  simplicity  7,  TI,  and  T2  will  be  retained,  but  it  must  be  remem- 
bered when  the  partial  derivatives  of  Ri,  2  are  taken  that  they  are 
functions  of  ii,  i*,  &>i,  and  ^2. 

As  a  consequence  of   (79),   (81),  and   (82),  the  perturbative 
function  can  be  written  in  the  form 


(84) 


,  2  = 


+  r22  —  2rir2  cos  (HI  —  u%  +  T2  — 
4rir2  sin  (ui  —  TI)  sin 


—  T2)  sin2  - 


+  r22  -  2nr2  cos  (ui  -  UZ  +  TZ-  TI) 
-  ^     cos  (ui  -  uz  +  T2  -  TI) 

—  2  sin  (ui  —  TI)  sin  (u^  —  T2)  sin2  -     . 

The  radii  ri  and  r2  are  independent  of  7.  The  second  factor  of 
the  first  term  of  the  right  member  of  this  equation  can  be  expanded 
by  the  binomial  theorem  into  an  absolutely  converging  power 

series  in  sin2  ~  so  long  as  the  numerical  value  of 


223] 


DEVELOPMENT  IN   POWERS   OF  61  AND 


409 


(85) 


sn 


n)  sin  (uz  —  r2)  sin2  - 


2  — 


r2    —     rir2  cos  («1  —  w2  +  r2  —  n) 
is  less  than  unity.     This  fraction  is  less  than,  or  at  most  equal  to, 

4-r,r«  sin2 

(86) 


4rir2  sin2  - 


(ri  - 


If  this  expression  is  less  than  unity  for  all  the  values  which  ri 
and  r2  can  take  in  the  given  ellipses  the  expansion  of  (84)  is  valid 
for  all  values  of  the  time.  In  the  case  of  the  major  planets  it  is 

always  very  small,  the  greatest  value  of  sin2  -  being  for  Mercury 

and  Mars,  0.0118.  In  the  perturbations  of  the  planetoids  by 
Jupiter  it  often  fails,  for  I  is  sometimes  of  considerable  magnitude 
while  r2  —  ri  may  become  very  small.  In  the  case  of  Mars  and 
Eros  r2  —  r\  may  actually  vanish  and  this  mode  of  development 
consequently  fails.  It  is  needless  to  say  that  it  is  not  generally 
applicable  in  the  cometary  orbits. 

In  those  cases  in  which  the  expansion  of  (84)  does  not  fail,  the 
expression  for  Rit  2  becomes 


(87)- 


Ri,  2=  +  [n2  +  r22  -  2nr2  cos  (MI  - 
r22  —  2/*ir2  cos  (u 


+  rz  -  n)]  * 
uz  +  r2  —  TI)]- 
X  2  sin  (MI  —  n)  sin  (uz  —  r2)  sin2 


cos   MI  - 


Tl 

rCOS 

r22 


X  6  sin2  (HI  —  n)  sin2  (uz  —  r2)  sin4  - 


—  u2  +  r2  — 


H  --  sin  (HI  —  TI)  sin  (uz  —  r2)  sin2  -  . 


7*2 


223.  (6)  Development  of  the  Coefficients  in  powers  of  ei  and  e2. 
The  radii  ri  and  r2  vary  from  ai(l  —  d)  and  a2(l  —  e2)  to  ai(l  + 
and  a2(l  +  e2)  respectively.     Let 


(88) 


f 

L 


p2). 


410  DEVELOPMENTS  IN  FOURIER  SERIES.  [224 

The  angles  HI  and  u2  are  expressed  in  terms  of  the  true  anomalies, 
Vi  and  v2,  and  the  elements  by  (82) .  The  true  anomalies  are  equal 
to  the  mean  anomalies  plus  the  equations  of  the  center,  which 
may  be  denoted  by  Wi  and  w2.  Let  li  and  12  represent  the  mean 
longitudes  counted  from  the  z-axis  [Fig.  (62)];  then 


(89) 

LW2   —    T2    =    12   —    &2   —    T2  +  W2. 

It  follows  from  (811)  that  Ri,  2  can  be  written  in  the  form 
Ri,  2  =  F[ai(l  +  PI),  a2(l  -f  p2)], 


where  F  is  a  homogeneous  function  of  ai  and  a2  of  degree  —  1. 
Therefore 

(90)  filtl  =         — 


The  right  member  of  this  equation  can  be  developed  by  Taylor's 
formula,  giving 


_  I  v(n  _  N  ,  PI  -  P2  ai  aF(oi,  qa) 

T-  -  S    P  (dl,  dz)    T  T~L  -   "i  --  5  - 

+  P2  I  1  +  p2   1         dai 

(  y  i  ) 


,    /PI  -P2\2  ai2    a2F(oi,  a2)  1 

r\    1+P2/     1    •    2    "       (to!2  '    J    ' 

The  expressions  (  ^       p2  J  can  be  developed  as  power  series  in 

Pi  and  p2.  But  in  Art.  100,  equation  (62)  ,  p  is  given  as  a  power  series 
in  e  whose  coefficients  are  cosines  of  multiples  of  the  mean  anomaly. 
On  making  these  expansions  and  substitutions  in  (91),  Ri,2  can 
be  arranged  as  a  power  series  in  e\  and  e2.  These  operations  are 
to  be  actually  performed  upon  the  separate  terms  of  the  series 
(87),  so  the  resulting  series  is  araanged  according  to  powers  of 

e\t  e2,  and  sin2  -  .     The  angles  Wi  and  w2  also  depend  upon  e\ 

and  e2  respectively,  but  their  developments  will  not  be  introduced 
until  after  the  next  step. 

224.  (c)  Developments  in  Fourier  Series.  The  first  term 
within  the  bracket  of  (91)  is  obtained  by  replacing  rj.  and  r2  by  ai 
and  a2  respectively  in  (87).  The  higher  terms  involve  the  deriva- 
tives of  the  first  with  respect  to  a\.  On  referring  to  the  explicit 
series  in  (87),  it  is  seen  that  the  development  of  the  expressions  of 
the  type 


224]  DEVELOPMENTS  IN   FOURIER  SERIES.  411 


v-\  _ 

2          2  —  —  —  , 


2  [ai2  +  a22  —  2aia2  cos  (ui  —  u2  +  r2  —  T       2 


where  *>  is  an  odd  integer,  must  be  considered. 

Let  HI  —  u2  +  r2  —  TI  =  ^.     It  is  known  from  the  theory  of 
Fourier  series  when  a\  and  a2  are  unequal,  as  is  assumed,  that 

_  V 

[a  i2  +  «22  —  2ai&2  cos  \f/]     2  can  be  developed  into  a  series  of  cosines 
of  multiples  of  \f/}  which  is  convergent  for  all  values  of  ^.     That  is, 


(92)     (a!a2)[ai2  +  a22  -  2aia2  cos  fl"*  =       V  £„<«  cos 


where  Bv™  =  BF<-«. 

The  coefficients  J3v(i)  are  of  course  given  by  Fourier's  integral 

1         /^7T  V-l  _V 

B/^  =  -  Jo     (aia2)  2  [Olz  +  a22  -  2aia2  cos  ^]   «  cos 


but  the  difficulty  of  finding  the  integral  makes  it  advisable  in  this 
particular  problem  to  proceed  otherwise. 

Let  z  =  e**~^,  where  e  represents  the  Napierian  base.     Then 

2  cos  \j/  =  z  +  z~1J       2  cos  i\l/  =  z{  +  z~*'. 

Suppose  a2  >  0,1  and  let  —  =  a;  then  (92)  becomes 
a2 

v-l 

(93)  —  (1  +  a2  -  2a  cos  ^)"5  =  i  E  ^v(i)  cos  i>. 

«2  ^  t=^oo 

Let 

(1  +  a2  -  2a  cos  i0~5  =  (1  -  a^)~^  (1  " 

therefore 

v-\ 

(94)  R«  =^-6/« 


Since  the  absolute  values  of  az  and  az~l  are  less  than  unity  for 

—  n  _v 

all  real  values  of  ^,  the  factors  (1  —  az)    *  and  (1  —  az~l)   *  can 

be  expanded  by  the  binomial  theorem  into  convergent  power 
series  in  az  and  car1.  The  coefficient  of  z1  in  the  product  of  these 
series  is  %by(i\  after  which  Bv(i)  is  obtained  from  (94).  The 
general  term  of  the  product  of  the  expansions  is  easily  found  to  be 


412  DEVELOPMENTS   IN  FOURIER   SERIES.  [224 


(95) 


1-2  (i  +  !)(;  +  2) 


In    this    manner  the  coefficients  of    p\JlpJ*  (  sin2-  )     are  de- 

\        */ 

veloped  in  Fourier  series  in  cos  i(^i  —  uz  +  r2  —  TI).  But  these 
functions  are  multiplied  by  the  factors  sin  (u\  —  TI)  sin  (u*  —  T2) 
raised  to  different  powers  [equation  (87)].  These  powers  of 
sines  are  to  be  reduced  to  sines  and  cosines  of  multiples  of  the 
arguments,  and  the  products  formed  with  cos  i(u\  —  u2  -f  r2  —  TI), 
and  the  reduction  again  made  to  sines  and  cosines  of  multiples 
of  arcs.  The  final  trigonometrical  terms  will  have  the  form 
cos  (  jiUi  +  jzuz  +  kiTi  +  &2r2),  where  ji,  jz,  ki,  and  &2  are  integers. 
As  a  consequence  of  (89)  this  expression  can  be  developed  into 


cos  (jii 
=  cos 

X  JCOS  (JiWi)  COS  (jzWz)   ~  sm  O'lWi)  sm 

X  {sin  (jiWi)  cos  0'2w2)  +  cos  (jiWi)  sin 
Since 

—  <o)  =  n\t  - 


the  first  factors  of  the  terms  in  the  right  member  of  this  equation 
are  independent  of  e\  and  e2.  Cos  (jiWi),  etc.,  are  to  be  expanded 
into  power  series  in  Wi  and  wz  by  the  usual  methods.  Now 
Wi  —  Vi  —  MI,  Wz  =  Vz  —  MZ,  and  these  quantities  were  developed 
into  power  series  in  e\  and  e2  [Art.  100,  eq.  (64)]  whose  coefficients 
were  Fourier  series  with  multiples  of  the  mean  anomaly  as  argu- 
ments. On  substituting  these  series  for  w\  and  w2  in  the  expansions 
of  the  second  factors  of  the  terms  of  the  right  member  of  (96) ,  and 
reducing  the  powers  of  sines  and  cosines  of  the  mean  anomaly  to 
sines  and  cosines  of  multiples  of  the  mean  anomaly,  and  multi- 
plying by  the  factors 

cos  (jili  -+•  jzlz  • 
and 


225]  PERIODIC  VARIATIONS.  413 

Sin   (jili   +  jzlz   -  jlftl   T  J2&2  +  fclTl  +  fe2T2), 


and  again  reducing  to  sines  and  cosines  of  multiples  of  the  argu- 
ments, the  expression  (96)  is  developed  as  a  power  series  in  e\ 
and  62  whose  coefficients  are  series  in  sines  and  cosines  of  sums  of 
multiples  of  liy  lz,  &i,  &2,  TI,  r2,  Mi,  M2.  But  MI  =  Zx  —  in,' 
M2  =  Z2  —  ?T2;  therefore  the  arguments  will  be  liy  12,  fti,  &2, 
TI,  T2,  TTi,  7r2,  where  TI  and  r2  are  functions  of  fti,  &2,  ii,  and  t'2 
denned  by  (83). 

When  the  several  expansions  and  reductions  which  have  been 
described  have  all  been  made,  Rit  2  will  be  developed  in  a  power 

series  in  e\,  e2,  and  sin2  •=  ,  the  coefficients  of  which  are  series  of 

sines  and  cosines  of  multiples  of  l\,  Z2,  &i,  &2,  n,  r2,  in,  7r2,  the 
coefficient  of  each  trigonometric  term  depending  upon  the  ratio 
of  the  major  semi-axes.  If  the  signs  of  £h,  &2,  in,  7r2,  TI,  T2, 
€1,  €2,  and  £  are  changed  the  value  of  Ri,  2,  as  denned  in  (84), 
obviously  is  unchanged;  therefore  the  expansion  in  question 
contains  only  cosines  of  the  argument.  Hence 


(97) 


,  z  =  2C  cos  D, 

+  kiTi  +  i 
C  =  /  (  01,  a2,  ei,  e2,  sin2  -  j  , 


in  which  ji,  •  •  • ,  A;2'  take  all  integral  values,  positive,  negative,  and 
zero,  the  summation  being  extended  over  all  of  these  terms. 

It  is  clear  from  the  foregoing  that  the  series  for  #1, 2  is  very 
complicated  and  that  much  labor  is  required  to  expand  it  in  any 
particular  case.  Leverrier  has  carried  out  the  literal  development 

of  all  terms  up  to  the  seventh  order  inclusive  in  e\t  ez,  sin2  - , 

it 

and  the  length  of  the  work  is  such  that  fifty-three  quarto  pages  of 
the  first  volume  of  the  Annales  de  I'Observatoire  de  Paris  are 
required  in  order  to  write  out  the  result. 

225.  Periodic  Variations.  It  follows  from  equations  (72)  and 
(97)  that  the  rates  of  change  of  the  elements  of  mi  are  given  by 


414 


PERIODIC   VARIATIONS. 


[225 


(98) 


dt 


VT= 


,, 


Sin 


dir\ 

~df 


j^          7      n 

_c062)_l_i 


ac 


n 

cos  D, 


l  —  ei2v^  f  7  /  ,  7   5n    ,  ,   ar2 

o 2^1  *i  +*13 ^^27— 

i2ei     ^^  I  d?ri         a?ri 


+ 


, 
w2  tan  — 


2     ^  f  ac  r    an,,  dT2l 

V  <  TT-  cos  D  -     ki  ^-  +  k2  rr- 

-  6l2  ^  I  aii  L    #ii        ^i  J 


cos 


The  perturbations  of  the  elements  of  the  orbit  of  mi  of  the  first 
order  with  respect  to  the  mass  w2  are  the  integrals  of  these  equa- 
tions regarding  the  elements  as  constants  in  the  right  members. 
Similar  terms  must  be  added  for  each  disturbing  planet. 

There  are  terms  in  Rit  2  of  three  classes:  (a)  those  in  which 
ji^i  +  jznz  is  distinct  from  zero  and  not  small;  (b)  those  in  which 
j\n\  +  J2n2  is  very  small,  but  distinct  from  zero;  and  (c)  those  in 
which  jini  +  J2n2  equals  zero.  Denote  the  fact  that  RI,  2  contains 
these  three  sorts  of  terms  by  writing 


225]  PERIODIC  VARIATIONS.  415 

Ri,  2  =  2Co  cos  DO  +  SCi  cos  DI  +  2C2  cos  D2, 

where  the  three  sums  in  the  right  member  include  these  three 
classes  of  terms  respectively.  Hence  the  perturbations  of  the 
elements  of  mi  by  w2  of  the  first  order  and  of  the  first  class  are 


(99) 


l  — 


{H/^ 
C/U 
dij 


sin  Do 


+  J2ft2 


sin  ii 

T    drz  1    Co  cos  Do 


6 12  sin  ii 


D 


w2  tan  — 


?7i2tan7r  f  _.~         .     T. 

_____?__  v1 1  —  -  sm 

o      1^ ^  ^-^    I      Qi' .     A 

,    dri    |    7    dr2  1  ^7o  COS  Do 


W2\l  — 


sin  Z) 


cos  -o 


nidi^J   jiWi  +J2n2 


W2  Vl  —  6i2  T->  f  7  i    ^TI    ,    T    dr2  1    Co  COS  D 

~ X        J     M   *      I       A*  L    f*A  _      _     \   

9  /  ^  i  /vi   ~r  "'i  »»        i    "/2  «       f  •          I 

nidi 26i  "   t  OTTi  OTTi  J  JiWi  -f-  J2? 

!70     sin  Do 


m2  tan  — 

fo  =  mai2Vf^l 
,    7    5r2 1   Co  COS  Do 


Do  1 

j2n2J 


,  1  -  Vl  -  ei2  ^  5C0     sin  Do 

+  m2 VI  -  ei2 -      —. 2. 

uiaizei       ^ 


sin  Dp 


416  LONG   PERIOD   VARIATIONS.   ,  [226 

These  terms  are  purely  periodic  with  periods  -  -  ~  —  ,  and 


constitute  the  periodic  variations.  Every  element  is  subject  to 
them,  depending  upon  an  infinity  of  such  terms  whose  periods 
are  different.  The  larger  jiWi  +  .7*2^2  is,  the  shorter  is  the  period 
of  the  term  and  in  general  the  smaller  is  its  coefficient. 

The  method  of  representing  the  motion  of  the  planets  by  a  series 
of  periodic  terms  is  somewhat  analogous  to  the  epicycloid  theory 
of  Ptolemy,  for  each  term  alone  is  equivalent  to  the  adding  of  a 
small  circular  motion  to  that  previously  existing.  This  theory  is 
more  complex  than  that  of  Ptolemy  in  that  it  adds  epicycloid 
upon  epicycloid  without  limit  ;'  it  is  simpler  than  that  of  Ptolemy 
in  that  it  flows  from  one  simple  principle,  the  law  of  gravitation. 

226.  Long  Period  Variations.  The  letters  ji  and  jz  represent 
all  positive  and  negative  integers  and  zero.  Therefore,  unless 
HI  and  nz  are  incommensurable  ji  and  j-2  exist  such  that  jini  + 
jznz  =  0,  where  ji  and  jz  are  not  zero.  But  then  D  is  a  constant 
and  the  integral  is  not  formed  this  way.  However,  whether  n\  and 
nz  are  incommensurable  or  not,  such  a  pair  of  numbers  can  be  found 
that  jiUi  -f  jznz  is  very  small.  The  corresponding  term  will  be 
large  unless  its  C  is  very  small.  It  is  shown  in  a  complete  dis- 
cussion of  the  development  of  Rit  2  that  the  order  of  C  in  e\,  ez, 

sin2  5  is  at  the  least  equal  to  the  numerical  value  of  ji  +  jz  (see 

Tisserand's  Mec.  C6L,  vol  i.,  p.  308).  Since  n\  and  n2  are  both 
positive,  one  of  the  numbers  ji,  jz  must  be  positive  and  the  other 
negative  in  order  that  the  sum  jiUi  +  j2nz  shall  be  small.  The 
more  nearly  equal  ji  and  jz  are  numerically  the  smaller  the  numeri- 
cal value  of  ji  +  jz  is,  and  consequently,  the  larger  C  will  be. 
When  the  mean  motions  of  the  two  planets  are  such  that  they  are 
nearly  commensurable  with  the  ratio  of  n\  to  n2  expressible  in 
small  integers,  then  large  terms  in  the  perturbations  will  arise 
from  the  presence  of  these  small  divisors.  The  period  of  such  a 

term  is  •=  -  -r—.  —  ,  which  is  very  great,  whence  the  appellation 

long  period.  These  terms  are  given  by  equations  of  the  same 
form  as  (99),  but  with  the  restriction  that  jini  +  jznz  shall  be 
very  small. 

Geometrically  considered,  the  condition  that  the  periods  shall 
be  nearly  commensurable  with  the  ratio  expressible  in  small 
integers  means  that  the  points  of  conjunction  occur  at  nearly  the 


227] 


SECULAR  VARIATIONS. 


417 


same  part  of  the  orbits  with  only  a  few  other  conjunctions  inter- 
vening. The  extreme  case  is  that  in  which  there  are  no  con- 
junctions intervening,  i.  e.,  when  ji  and  j2  differ  in  numerical  value 
by  unity. 

The  mean  motions  of  Jupiter  and  Saturn  are  nearly  in  the  ratio 
of  five  to  two.  Consequently  ji  =  2,  j2  —  —  5  gives  a  long 
period  term,  and  the  order  of  the  coefficient  C  is  the  absolute 
value  of  2  —  5,  or  3.  The  cause  of  the  long  period  inequality  of 
Jupiter  and  Saturn  was  discovered  by  Laplace  in  1784  in  com- 
puting the  perturbations  of  the  third  order  in  ei  and  e2.  The 
length  of  the  period  in  the  case  of  these  two  planets  is  about  850 
years. 

227.  Secular  Variations.  The  expression  D  is  independent 
of  the  time  for  all  of  those  terms  in  which  ji  =  jz  =  0.  The 
partial  derivatives  of  D  with  respect  to  the  elements  are  also 
independent  of  the  time;  hence,  on  taking  these  terms  of  (98)  and 
integrating,  it  is  found  that 


(100) 


ra2 


sn 


Z          2 


—  ra2 


VI  — 


sn 


2Vi  - 


tan  I 


nidi 


*vr^ 


--  cos  L>2 


[A:^+ /c2^?  J  C2  sin 


28 


418 


(100)  1 


SECULAR  VARIATIONS. 

o, 


[227 


w2  tan  — 

Zi 


dC, 


-j-  m2  Vl  - 


2w2 


1    -  Vl   - 


X 


It  follows  that  there  are  no  secular  terms  of  this  type  of  the  first 
order  with  respect  to  the  masses  in  the  perturbations  of  a.  This 
constitutes  the  first  theorem  on  the  stability  of  the  solar  "system. 
It  was  proved  up  to  the  second  powers  of  the  eccentricities  by 
Laplace  in  1773,*  when  he  was  but  twenty-four  years  of  age,  in 
a  memoir  upon  the  mutual  perturbations  of  Jupiter  and  Saturn; 
it  was  shown  by  Lagrange  in  1776  that  it  is  true  for  all  powers  of 
the  eccentricities.f  It  was  proved  by  Poisson  in  1809  that  there 
are  no  secular  terms  in  a  in  the  perturbations  of  the  second  order 
with  respect  to  the  masses,  but  that  there  are  terms  of  the  type 
t  cos  D,  where  D  contains  the  time  4  Terms  of  this  type  are 
commonly  called  Poisson  terms. 

All  of  the  elements  except  a  have  secular  terms.  It  appears 
to  have  been  supposed  that  the  secular  terms,  which  apparently 
cause  the  elements  to  change  without  limit,  alone  prevent  the  use 
of  equations  (72)  for  computing  the  perturbations  for  any  time 
however  great.  Many  methods  of  computing  perturbations  have 
been  devised  in  order  to  avoid  the  appearance  of  secular  terms; 
yet  it  is  clear  that,  whether  or  not  terms  proportional  to  the  time 

*  Memoir  presented  to  the  Paris  Academy  of  Sciences. 
t  Memoirs  of  the  Berlin  Academy,  1776. 
t  Journal  de  I'Ecole  Poly  technique,  vol.  xv. 


228]  TERMS   OF  THE   SECOND   ORDER.  419 

appear,  the  method  is  strictly  valid  for  only  those  values  of  the 
time  for  which  the  series  (20)  of  Art.  207  are  convergent. 

Secular  terms  may  enter  in  another  way,  usually  not  considered. 
If  jini  +  J2n2  =  0  with  ji  =|=  0,  j2  =t=  0,  D  is  independent  of  the 
time  and  the  corresponding  terms  are  secular.  In  this  case  D  is 
not  independent  of  ei  and  there  will  be  secular  terms  in  the  per- 
turbations of  a.  As  has  been  remarked,  this  condition  will  always 
be  fulfilled  by  an  infinity  of  values  of  j\  and  j2  if  n\  and  n2  are  not 
incommensurable.  But  it  is  impossible  to  determine  from  obser- 
vations whether  or  not  ni  and  n2  are  incommensurable,  for  there 
is  always  a  limit  to  the  accuracy  with  which  observations  can  be 
made,  and  within  this  limit  there  exist  infinitely  many  com- 
mensurable and  incommensurable  numbers.  There  is  as  much 
reason,  therefore,  to  say  that  secular  terms  in  a  of  this  type  exist 
as  that  they  do  not.  However,  they  are  of  no  practical  im- 
portance because  the  ratio  of  HI  to  n2  cannot  be  expressed  in  small 
integers,  and  the  coefficients  of  these  terms,  if  they  do  exist,  are 
so  small  that  they  are  not  sensible  for  such  values  of  the  time  as  are 
ordinarily  used. 

228.  Terms  of  the  Second  Order  with  Respect  to  the  Masses. 

The  terms  of  the  second  order  are  defined  by  equations  (29), 
Art.  210.  The  right  members  of  these  equations  are  the  products 
of  the  partial  derivatives,  with  respect  to  the  elements,  of  the  right 
members  which  occur  in  the  terms  of  the  first  order,  and  the 
perturbations  of  the  first  order  of  the  corresponding  elements. 
Thus,  the  second  order  perturbations  of  the  node  are  determined 
by  the  equations 


dt  Uiai2  Vl  —  6i2  sin 

(101) 

~!^< 


Vl  -  ei2  sin  ii  i   <« 
where  Si  and  s2  represent  the  elements  of  the  orbits  of  mi  and  m2 

*\2  D 

respectively.     The  partial  derivative      .  *' 2  is  a  sum  of  periodic 

C7  2/iC/u  ]_ 

and  constant  terms;  s/0-^  and  s2(1'0)  are  sums  of  periodic  terms 
and  terms  containing  the  time  to  the  first  degree  as  a  factor.     The 

products        1;  2  si(0>  1}  and     .  *' 2  s2(1-0)  therefore  contain  terms  of 


420        LAGRANGE'S  TREATMENT  OF  SECULAR  VARIATIONS.       [229 

four  types:   (a)  sm  D,  where  D  contains  the  time;   (b)  ts™  D] 
cos  cos 

(c)  ^m  Z>2,  where  D2  is  independent  of  the  time;  and  (d)  t  ^  D2. 
The  integrals  of  these  four  types  are  respectively: 

—  cos  T}  —  cos  ft  sin  ^ 

sin  sin  cos 


+  jzn 


Therefore,  the  perturbations  of  the  second  order  with  respect  to 
the  masses  have  purely  periodic  terms;  Poisson  terms,  or  terms 
in  which  the  trigonometric  terms  are  multiplied  by  the  time; 
secular  terms  where  the  time  occurs  to  the  first  degree;  and  secular 
terms  where  the  time  occurs  to  the  second  degree.  This  is  true 
for  all  of  the  elements  except  the  major  semi-axis,  in  the  case  of 
which  the  coefficients  of  the  terms  of  the  third  and  fourth  types 
are  zero,  as  Poisson  first  proved. 

In  the  terms  of  the  third  order  with  respect  to  the  masses  there 
are  secular  terms  in  the  perturbations  of  all  the  elements  except 
d,  which  are  proportional  to  the  third  power  of  the  time,  and  so  on. 

229.  Lagrange's  Treatment  of  the  Secular  Variations.  The 
presence  of  the  secular  terms  in  the  expressions  for  the  elements 
seems  to  indicate  that,  if  it  is  assumed  that  the  series  represent 
the  elements  for  all  values  of  the  time,  then  the  elements  change 
without  limit  with  the  time.  But  this  conclusion  is  by  no  means 
necessarily  true.  For  example,  consider  the  function 


/3 

(102)  sin  (cmt)  =  cmt  -    ~.  -\  ----  , 

o! 

where  c  is  a  constant  and  m  a  very  small  factor  which  may  take  the 
place  of  a  mass.  The  series  in  the  right  member  converges  for 
all  values  of  t.  This  function  is  never  greater  than  unity  for  any 
value  of  the  time;  yet  if  its  expansion  in  powers  of  m  were  given, 
and  if  the  first  few  terms  were  considered  without  the  law  of  the 
coefficients  being  known,  it  might  seem  that  the  series  represents 
a  function  which  increases  indefinitely  in  numerical  value  with 
the  time. 

On  following  out  the  idea  that  the  secular  terms  may  be  ex- 


229]       LAGRANGE'S  TREATMENT  OF  SECULAR  VARIATIONS.        421 


pansions  of  functions  which  are  always  finite,  Lagrange  has  shown 
(see  Collected  Works,  vols.  v.  and  vi.),  under  certain  assumptions 
which  have  not  been  logically  justified,  that  the  secular  terms  are 
in  reality  the  expansions  of  periodic  terms  of  very  long  period. 
These  terms  differ  from  the  long  period  variations  (Art.  226)  in 
that  they  come  from  the  small  uncompensated  parts  of  the  periodic 
variations,  instead  of  directly  from  special  conditions  of  con- 
junctions. As  a  rule  these  terms  are  very  small,  and  their  periods 
are  much  longer  than  those  of  the  sensible  long  period  terms.  It 
will  not  be  possible  to  give  here  more  than  a  very  general  idea  of 
the  method  of  Lagrange. 

The  first  step  in  the  method  of  Lagrange  is  a  transformation  of 
variables  by  the  equations 


(103) 
and 
(104) 


=  €j  Sin  7T/, 
=   6j  COS  7T/, 

Pi  =  tan  ij  sin  £< 
#/  =  tan  ij  cos  £ 


where  eh  TT/,  etc.,  are  the  elements  of  the  orbit  of  mh  and  lj  is  a 
new  variable  not  to  be  confused  with  the  mean  longitude.  These 
transformations  are  to  be  made  simultaneously  in  the  elements  of 
the  orbits  of  all  of  the  planets.  The  elements  a/  and  e/  remain 
without  transformation.  On  omitting  the  subscripts,  it  is  found 
from  (103)  and  (104)  that 


(105) 


'  dh 

dir  .     .       de 

dl 

=  -  e  sin  • 

w  — 

+  cosx(fe 

dR 

de 

_  dR  dh 
~  dh  de 

dR 

az  _   . 

de  ~ 

dR  .            dR 

7T  -TT-  +   COS  T  -77 
6/1                             6t 

[; 

dR 

_dRdh 

dR 

dl  _ 

a^ 

dR 

dir 

~  dh  dir 

dl 

dir 

18  ^aF" 

dl 

Tt 

=  -f-  tan  i 

cos 

ad& 
1  rf< 

^<K 

sec2  1  sin  ft  37  , 
at 

dq 
_   dt 

=  —  tan  i 

sin 

nd&    . 
0  d<    " 

-.in 

sec2  z  cos  &  -T.  , 
at 

422        LAGRANGE'S  TREATMENT  OF  SECULAR  VARIATIONS.       [229 


dR  =  dR  dp       dR  dq 
d&  ~  dp  d&        dq  d& 

^dR  .  . 

=  tan  i  cos  Q>  -  --  tan  i  sin 

dp 


=  , 

di  ~  dp  di        dq  di 

dR 


dK 
dq' 


dR 


=  sec2  i  sin  Q>  - — \-  sec2  i  cos  &  -r— . 


dp 

Then  it  follows  from  (72)  that 
e&      m2  Vl  -  h2  -  I2 


dq 


(106)   H 


na 


m2  Vl  -  h2  -  I2 


no 


mjtan- 


Vl  -hz-l2 


na 


-  m2  Vl  -  /t2  - 


2 


m2  Vl  -  A2  - 


dR 


na 


m2  A  tan  « 


dR 


na2  Vl  -  /i2  - 


cos 


3 


2na2  Vl  -  /i2  -  /2  cos  i  cos2  » 


+ 


—  1, 
ae  J 


na 


2  Vl  -  /I2  -  Z2  cos3  i  3? 


ra#+a£j 

t-  L  a?r         6e  J 


2na2  Vl  —  h2  —  I2  cos  i  cos2  - 

2i 

On  developing  the  right  members  of  these  equations  and  neglecting 
all  terms  of  degree  higher  than  the  first*  in  h,  I,  p,  and  q,  these 

*  The  terms  of  order  higher  than  the  first  are  neglected  throughout  in  a 
later  step  in  the  method. 


229]       LAGRANGE'S  TREATMENT  OF  SECULAR  VARIATIONS.        423 

equations  reduce  to 

dh 


(107) 


~dt 


no?  dh  ' 
mzdR 


dp  _ 

dt  "  *"  no?  dq  ' 

dq  _  m2  dR 

~dt~  ~  na2  ~dp  ' 


The  terms  which  involve  the  derivative  of  R  with  respect  to  e,  i, 
and  TT  do  not  appear  in  these  equations  because  they  involve  h,  I, 
p,  or  q  as  a  factor.  This  fact  follows  from  the  properties  of  C 
given  in  Art.  226  and  the  form  of  equations  (103)  and  (104). 

Each  perturbing  planet  contributes  terms  in  the  right  members 
of  equations  (107)  similar  to  the  ones  written  which  come  from  ra2. 
These  differential  equations  are  not  strictly  correct,  since  the 
first  approximation  has  already  been  made  in  neglecting  the  higher 
powers  of  the  variables. 

The  second  step  is  in  the  method  of  treating  the  differential 
equations.  The  expansions  of  the  Ri,  /  contain  certain  terms 
which  are  independent  of  the  time,  which  in  the  ordinary  method 
give  rise  to  the  secular  terms.  Let  Rwi,  /  represent  these  terms. 
Lagrange  then  treated  the  differential  equations  by  neglecting  the 
periodic  terms  in  Rit  ,-,  and  writing 


(i  =1,   ••-,  n]  j  4=  i), 


(108) 


The  values  of  hi,  Z»,  p»,  and  g»  determined  from  equations 
(108)  are  used  instead  of  the  secular  terms  obtained  by  the 
method  of  Art.  227.  The  process  of  breaking  up  a  differential 
equation  in  this  manner  is  not  permissible  except  as  a  first  approxi- 
mation, and  any  conclusions  based  on  it  are  open  to  suspicion. 


424 


LAGEANGE  S  TREATMENT  OF  SECULAR  VARIATIONS. 


[229 


In  spite  of  the  logical  defects  of  the  method  and  the  fact  that  it 
cannot  be  generally  applied,  there  is  little  doubt  that  in  the 
present  case  it  gives  an  accurate  idea  of  the  actual  manner  in  which 
the  elements  vary. 

The  right  members  of  equations  (108)  are  expanded  in  powers  of 
hi,  li,  p^  and  #»,  and  all  of  the  terms  except  those  of  the  first  degree 
are  neglected;  consequently  the  terms  omitted  in  (107)  would 
have  disappeared  here  if  they  had  been  retained  up  to  this  point. 
The  system  becomes  linear,  and  the  detailed  discussion  of  the 
R it  j  shows  that  it  is  homogeneous,  giving  equations  of  the  form 


(109) 


dfe 

dt 


dh» 


and  a  similar  system  of  equations  in  the  PJ  and  the  <?/. 

The  coefficients  ct/  depend  only  on  the  major  axes  (the  e/  not 
appearing  in  the  secular  terms)  which  are  considered  as  being 
constants,  since  the  major  axes  have  no  secular  terms  in  the 
perturbations  of  the  first  and  second  orders  with  respect  to  the 
masses.  It  is  to  be  noted  here  that  the  assumption  that  the  c^ 
are  constants  is  not  strictly  true  because  the  major  axes  have 
periodic  perturbations  which  may  be  of  considerable  magnitude. 

When  these  linear  equations  are  solved  by  the  method  used  in 
Art.  160,  the  values  of  the  variables  are  found  in  the  form 


(110) 


Pi  - 


230]     PERTURBATIONS  BY  MECHANICAL  QUADRATURES.      425 

where  the  H^,  L^,  Pih  and  Qij,  are  constants  depending  upon  the 
initial  conditions.  A  detailed  discussion  shows  that  the  X/  and  /*/ 
are  all  pure  imaginaries  with  very  small  absolute  values;  there- 
fore the  hi,  It,  pi,  and  gt-  oscillate  around  mean  values  with  very 
long  periods.  Or,  since  the  e,-  and  tan  t,-  are  expressible  as  the 
sums  of  squares  of  the  h,,  lh  ph  and  qh  it  follows  that  they  also 
perform  small  oscillations  with  long  periods;  for  example,  the 
eccentricity  of  the  earth's  orbit  is  now  decreasing  and  will  continue 
to  decrease  for  about  24,000  years. 

Equations  (109)  admit  integrals  first  found  by  Laplace  in  1784, 
which  lead  practically  to  the  same  theorem.     They  are 

mjn,jaf(hf  +  If)  =  Constant  =  C, 
(111) 


or,  because  of  (103)  and  (104), 

mjrijafef  =  C, 

£_   fij^-.   C  ^   <5^^ 

jUjaj2  tan2  ty  =  C', 

where  n,-  is  the  mean  motion  of  my.  The  constants  C  and  C"  as 
determined  by  the  initial  conditions  are  very  small,  and  since  the 
left  members  of  (112)  are  made  up  of  positive  terms  alone,  no  e, 
or  ij  can  ever  become  very  great.  There  might  be  an  exception 
if  the  corresponding  my  were  very  small  compared  to  the  others. 

Equations  (112)  give  the  celebrated  theorems  of  Laplace  that 
the  eccentricities  and  inclinations  cannot  vary  except  within  very 
narrow  limits.  Although  the  demonstration  lacks  complete  rigor, 
yet  the  results  must  be  considered  as  remarkable  and  significant. 
Equations  (112)  do  not  give  the  periods  and  amplitudes  of  the 
oscillations  as  do  equations  (110). 

230.  Computation  of  Perturbations  by  Mechanical  Quadratures. 

If  the  second  term  of  the  second  factor  of  (84)  in  absolute  value  is 
greater  than  unity,  the  series  (87)  does  not  converge  and  cannot 
be  used  in  computing  perturbations.  The  expansions  may  fail 
because  r\  and  r2  are  very  nearly  equal;  or,  sometimes  when  they 
are  not  nearly  equal,  because  /  is  large.  In  the  latter  case 


426  PERTURBATIONS   BY   MECHANICAL   QUADRATURES.  [230 

another  mode  of  expansion  sometimes  can  be  employed,  *  but  there 
are  cases  in  which  neither  method  leads  to  valid  results.  They 
both  fail  if  the  two  orbits  placed  in  the  same  plane  would  intersect, 
for  in  this  case 

r2i,  2  =  ri2  +  r22  -  2rlr2  cos  (ui  -  u2  +  r2  -  n), 

would  vanish  when  the  two  bodies  arrive  at  a  point  of  inter- 
section of  their  orbits  at  the  same  time.  Unless  the  periods  are 
commensurable  in  a  special  way  this  would  always  happen.  Of 
course,  it  is  not  necessary  that  ri,  2  should  actually  vanish  in 
order  that  the  expansion  of  (84)  should  fail  to  converge. 

Perturbations  can  be  computed  by  the  method  of  mechanical 
quadratures  without  expanding  the  perturbative  function  explicitly 
in  terms  of  the  time.  Consequently,  this  method  can  be  used  in 
computing  the  disturbing  effects  of  planets  on  comets  and  in  other 
cases  where  the  expansion  of  R  i,  2  fails  altogether  or  converges 
slowly.  Let  s  represent  an  element  of  the  orbit  of  Wi;  then 
equations  (77)  can  be  written  in  the  form 


dt 
and  the  perturbations  of  the  first  order  in  the  interval  tn  —  tQ  are 

(113)  8   =   So  + 


where  s0  is  the  value  of  s  at  t  =  to. 

The  only  difficulty  in  computing  perturbations  is  in  forming  the 
integrals  indicated  in  (113).  When  the  perturbative  function  can- 
not be  expanded  explicitly  in  terms  of  t  the  primitive  of  the 
function  fa(t)  cannot  be  found.  But  in  any  case  the  values  of 
f»(t)  can  be  found  for  any  values  of  t,  and  from  the  values  of  fs(t) 
for  special  values  of  t  an  approximation  to  the  integral  can  be 
obtained.  Geometrically  considered,  the  integral  (113)  is  the 
area  comprised  between  the  £-axis  and  the  curve  /  =  f,(t)  and  the 
ordinates  t0  and  tn.  An  approximate  value  of  the  integral  is 

8   =    SO  +/.(«0)(<1   ~   <o)    +/.(«l)(«2   ~   *l)    +    '  "-\-fs(tn-l)(tn    ~   <n-l). 

The  intervals  ti  —  t0,  tz  —  ti,  •  -  -  ,  tn  —  tn_i  can  be  taken  so  small 
that  the  approximation  will  be  as  close  as  may  be  desired. 

Another  method  of  obtaining  an  approximate  value  of  the  inte- 
*  Tisserand,  Mecanique  Celeste,  vol.  i.,  chap,  xxvin. 


230]     PERTURBATIONS  BY  MECHANICAL  QUADRATURES.      427 

gral  is  to  replace  the  curve  /8(0,  whose  explicit  value  in  convenient 
form  may  not  be  obtainable,  by  a  polynomial  curve  of  the  nth 
degree  which  agrees  in  value  with  fs(i)  at  t  =  to,  ti,  •  •  • ,  tn.  The 
equation  of  this  polynomial  is 

f. 


(«-«l)(*-*2)  ••• 

(t-Q    ffn 

'    (*o-  *i)(«o-  fc)  '•• 
(«-*o)(*-«  IV 

«o  -  U  /s(iu) 

'      (tl   ~    to)(t!   -    t2)    '" 

(4            4  \  Ja^l) 

(tl  —   tn) 

^ 


Since  there  is  no  trouble  in  forming  the  integral  of  a  polynomial 
there  is  no  trouble  in  computing  the  perturbation  of  s  for  the  in- 
terval tn  —  to.  If  the  value  of  the  function  fs(t)  is  not  changing 
very  rapidly  or  irregularly,  its  representation  by  a  polynomial  is 
very  exact  provided  the  intervals  ti  —  to,  -  •  •  ,  tn  —  tn-\  are  not 
too  great. 

However,  the  area  between  the  polynomial,  the  £-axis,  and  the 
limiting  ordinates  is  not  the  best  approximation  to  the  value  of 
the  integral  that  can  be  obtained  from  the  values  of  fa(t)  at  t0, 
•  •-  •  ,  t^  The  values  of  the  function  give  information  respecting 
the  nature  of  the  curvature  of  the  curve  between  the  ordinates 
(this  being  true,  of  course,  only  because  the  function  f,(t)  is  a 
regular  function  of  t),  and  corrections  of  the  area  due  to  these 
curvatures  can  easily  be  made.  Ordinarily  they  would  involve  the 
derivatives  of  fs(t)  at  £o,  •  •  •  ,  tn,  which  would  require  a  vast  amount 
of  labor  to  compute;  but  the  derivatives  can  be  expressed  with 
sufficient  approximation  in  terms  of  the  successive  differences  of 
the  function,  and  the  differences  are  obtained  directly  from  the 
tabular  values  by  simple  subtraction.  The  derivation  of  the 
most  convenient  explicit  formulas  is  a  lengthy  matter  and  must 
be  omitted.* 

Suppose  the  computation  of  the  integrals  from  the  values  of 
fa(t)  at  t  =  to,  •  •  •  ,  tn  has  not  given  results  which  are  sufficiently 
exact.  More  exact  ones  can  be  obtained  by  dividing  the  interval 
tn  —  to  into  a  greater  number  of  sub-intervals.  A  little  experience 
usually  makes  it  unnecessary  to  subdivide  the  intervals  first  chosen. 

*  See  Tisserand's  Mecanique  Ctleste,  vol.  iv.,  chaps,  x.  and  XL;  and  Char« 
lier's  Mechanik  des  Himmels,  vol.  n.,  chap.  1. 


428  PERTURBATIONS    BY  MECHANICAL   QUADRATURES.  [230 

There  is  a  second  reason  why  the  results  obtained  by  mechanical 
quadratures  may  not  be  sufficiently  exact.  It  has  so  far  been 
assumed  that/s(0  is  a  function  of  t  alone;  or,  in  other  words,  that 
the  elements  of  the  orbits  on  which  it  depends  are  constants. 
This  is*the  assumption  in  computing  perturbations  of  the  first 
order.  If  it  is  not  exact  enough,  new  values  of  /8(£i),  •  •  •,  fs(tn) 
can  be  computed,  on  using  in  them  the  respective  values  of 
the  elements  s  which  were  found  by  the  first  integration.  From 
the  new  values  of  fs(ti),  •  •  •,  fs(tn)  a  more  approximate  value  of 
the  integral  can  be  obtained.  Unless  the  interval  tn  —  t0  is  too 
great  this  process  converges  and  the  integral  can  be  found  with 
any  desired  degree  of  approximation,  because  this  method  is 
simply  Picard's  method  of  successive  approximations  whose 
validity  has  been  established.*  In  practice  it  is  always  advisable 
to  choose  the  interval  tn  —  t0  so  short  that  no  repetition  of  the 
computation  with  improved  values  of  the  function  at  the  ends  of 
the  sub-intervals  will  be  required.  At  each  new  stage  of  the  inte- 
gration the  values  of  the  elements  at  the  end  of  the  preceding 
step  are  employed.  It  follows  that  the  method,  as  just  explained, 
enables  one  to  compute  not  only  the  perturbations  of  the  first  order, 
but  perturbations  of  all  orders  except  for  the  limitations  that 
the  intervals  cannot  be  taken  indefinitely  small  and  the  compu- 
tation cannot  be  made  with  indefinitely  many  places. 

The  process  of  computing  perturbations  by  the  method  of 
mechanical  quadratures,  as  compared  with  that  of  using  the 
expanded  form  of  the  perturbative  function,  has  its  advantages 
and  its  disadvantages.  It  is  an  advantage  that  in  employing 
mechanical  quadratures  it  is  not  necessary  to  express  the  per- 
turbing forces  explicitly  in  terms  of  the  elements  and  the  time. 
This  is  sometimes  of  great  importance,  for,  in  cases  where  the 
eccentricities  and  inclinations  are  large,  as  in  some  of  the  asteroid 
orbits,  these  expressions,  which  are  series,  are  very  slowly  con- 
vergent; and  in  the  case  of  orbits  whose  eccentricities  exceed 
0.6627,  or  of  orbits  which  have  any  radius  of  one  equal  to  any 
radius  of  the  other  the  series  are  divergent  and  cannot  be  used. 
The  method  of  mechanical  quadratures  is  equally  applicable  to 
all  kinds  of  orbits,  the  only  restriction  being  that  the  intervals 
shall  be  taken  sufficiently  short.  It  is  the  method  actually  em- 
ployed, in  one  of  its  many  forms,  in  computing  the  perturbations 
of  the  orbits  of  comets. 

*  Picard's  Traite  d' Analyse,  vol.  n.,  chap.  XL,  section  2. 


231]  GENERAL  REFLECTIONS.  429 

The  disadvantages  are  that,  in  order  to  find  by  mechanical 
quadratures  the  values  of  the  elements  at  any  particular  time, 
it  is  necessary  to  compute  them  at  all  of  the  intermediate  epochs. 
Being  purely  numerical,  it  throws  no  light  whatever  on  the  general 
character  of  perturbations,  and  leads  to  no  general  theorems 
regarding  the  stability  of  a  system.  These  are  questions  of 
great  interest,  and  some  of  the  most  brilliant  discoveries  in  Ce- 
lestial Mechanics  have  been  made  respecting  them. 

231.  General  Reflections.  Astronomy  is  the  oldest  science 
and  in  a  certain  sense  the  parent  of  all  the  others.  The  relatively 
simple  and  regularly  recurring  celestial  phenomena  first  taught 
men,  in  the  days  of  the  ancient  Greeks,  that  Nature  is  systematic 
and  orderly.  The  importance  of  this  lesson  can  be  inferred  from 
the  fact  that  it  is  the  foundation  on  which  all  science  is  based. 
For  a  long  time  progress  was  painfully  slow.  Centuries  of  obser- 
vations and  attempts  at  theories  for  explaining  them  were  neces- 
sary before  it  was  finally  possible  for  Kepler  to  derive  the  laws 
which  are  a  first  approximation  to  the  description  of  the  way  in 
which  the  planets  move.  The  wonder  is  that,  in  spite  of  the 
distractions  of  the  constant  struggles  incident  to  an  unstable 
social  order,  there  should  have  been  so  many  men  who  found  their 
greatest  pleasure  in  patiently  making  the  laborious  observations 
which  were  necessary  to  establish  the  laws  of  the  celestial  motions. 

The  work  of  Kepler  closed  the  preliminary  epoch  of  two  thousand 
years,  or  more,  and  the  brilliant  discoveries  of  Newton  opened 
another.  The  invention  of  the  Calculus  by  Newton  and  Leibnitz 
furnished  for  the  first  time  mathematical  machinery  which  was 
at  all  suitable  for  grappling  with  such  difficult  problems  as  the 
disturbing  effects  of  the  sun  on  the  motion  of  the  moon,  or  the 
mutual  perturbations  of  the  planets.  It  was  fortunate  that  the 
telescope  was  invented  about  the  same  time;  for,  without  its  use, 
it  would  not  have  been  possible  to  have  made  the  accurate  obser- 
vations which  furnished  the  numerical  data  for  the  mathematical 
theories  and  by  which  they  were  tested.  The  history  of  Celestial 
Mechanics  during  the  eighteenth  century  is  one  of  a  continuous 
series  of  triumphs.  The  analytical  foundations  laid  by  Clairaut, 
d'Alembert,  and  Euler  formed  the  basis  for  the  splendid  achieve- 
ments of  Lagrange  and  Laplace.  Their  successors  in  the  nine- 
teenth century  pushed  forward,  by  the  same  methods  on  the 
whole,  the  theories  of  the  motions  of  the  moon  and  planets  to 
higher  orders  of  approximation  and  compared  them  with  more 


430  PROBLEMS. 

and  better  observations.  In  this  connection  the  names  of  Lever- 
rier,  Delaunay,  Hansen,  and  Newcomb  will  be  especially  remem- 
bered. Near  the  close  of  the  nineteenth  century  a  third  epoch 
was  entered.  It  is  distinguished  by  new  points  of  view  and  new 
methods  which,  in  power  and  mathematical  rigor,  enormously 
surpass  all  those  used  before.  It  was  inaugurated  by  Hill  in  his 
Researches  on  the  Lunar  Theory,  but  owes  most  to  the  brilliant  con- 
tributions of  Poincare  to  the  Problem  of  Three  Bodies. 

At  the  present  time  Celestial  Mechanics  is  entitled  to  be  regarded 
as  the  most  perfect  science  and  one  of  the  most  splendid  achieve- 
ments of  the  human  mind.  No  other  science  is  based  on  so  many 
observations  extending  over  so  long  a  time.  In  no  other  science 
is  it  possible  to  test  so  critically  its  conclusions,  and  in  no  other 
are  theory  and  experience  in  so  perfect  accord.  There  are  thou- 
sands of  small  deviations  from  conic  section  motion  in  the  orbits 
of  the  planets,  satellites,  and  comets  where  theory  and  the  obser- 
vations exactly  agree,  while  the  only  unexplained  irregularities 
(probably  due  to  unknown  forces)  are  a  very  few  small  ones  in 
the  motion  of  the  moon  and  the  motion  of  the  perihelion  of  the 
orbit  of  Mercury.  Over  and  over  again  theory  has  outrun  practise 
and  indicated  the  existence  of  peculiarities  of  motion  which  had 
not  yet  been  derived  from  observations.  Its  perfection  during 
the  time  covered  by  experience  inspires  confidence  in  following  it 
back  into  the  past  to  a  time  before  observations  began,  and  into 
the  future  to  a  time  when  perhaps  they  shall  have  ceased.  As 
the  telescope  has  brought  within  the  range  of  the  eye  of  man  the 
wonders  of  an  enormous  space,  so  Celestial  Mechanics  has  brought 
within  reach  of  his  reason  the  no  lesser  wonders  of  a  correspond- 
ingly enormous  time.  It  is  not  to  be  marveled  at  that  he  finds 
profound  satisfaction  in  a  domain  where  he  is  largely  freed  from 
the  restrictions  of  both  space  and  time. 

XXVII.     PROBLEMS. 

1.  Suppose  (a)  that  R i,  2  is  large  and  nearly  constant;    (6)  that  R\,z  is 
large  and  changing  rapidly;  (c)  that  Ri,  2  is  small  and  nearly  constant.     If  the 
perturbations   are   computed   by   mechanical   quadratures   how   should   the 
/„  —  to  be  chosen  relatively  in  the  three  cases,  and  how  should  the  numbers  of 
subdivisions  of  tn  —  t0  compare? 

2.  The  perturbative  function  involves  the  reciprocal  of  the  distance  from 
the  disturbing  to  the  disturbed  planets.     This  is  called  the  principal  part  and 
gives  the  most  difficulty  in  the  development.     How  many  separate  reciprocal 


HISTORICAL   SKETCH.  431 

distances  must  be  developed  in  order  to  compute,  in  a  system  of  one  sun  and 
n  planets,  (a)  the  perturbations  of  the  first  order  of  one  planet;  (6)  the  per- 
turbations of  the  first  order  of  two  planets;  (c)  the  perturbations  of  the  second 
order  of  one  planet;  and  (d)  the  perturbations  of  the  third  order  of  one  planet? 

3.  What  simplifications  would  there  be  in  the  development  of  the  per- 
turbative  function  if  the  mutual  inclinations  of  the  orbits  were  zero,  and  if 
the  orbits  were  circles? 

4.  What  sorts  of  terms  will  in  general  appear  in  perturbations  of  the  third 
order  with  respect  to  the  masses? 


HISTORICAL  SKETCH  AND   BIBLIOGRAPHY. 

The  theory  of  perturbations,  as  applied  to  the  Lunar  Theory,  was  developed 
from  the  geometrical  standpoint  by  Newton.  The  memoirs  of  Clairaut  and 
D'Alembert  in  1747  contained  important  advances,  making  the  solutions 
depend  upon  the  integration  of  the  differential  equations  in  series.  Clairaut 
soon  had  occasion  to  apply  his  processes  of  integration  to  the  perturbations 
of  Halley's  comet  by  the  planets  Jupiter  and  Saturn.  This  comet  had  been 
observed  in  1531,  1607,  and  1682.  If  its  period  were  constant  it  would  pass 
the  perihelion  again  about  the  middle  of  1759.  Clairaut  computed  the 
perturbations  due  to  the  attractions  of  Jupiter  and  Saturn,  and  predicted  that 
the  perihelion  passage  would  be  April  13,  1759.  He  remarked  that  the  time 
was  uncertain  to  the  extent  of  a  month  because  of  the  uncertainties  in  the 
masses  of  Jupiter  and  Saturn  and  the  possibility  of  perturbations  from  un- 
known planets  beyond  these  two.  The  comet  passed  the  perihelion  March  13, 
giving  a  striking  proof  of  the  value  of  Clairaut's  methods. 

The  theory  of  the  perturbations  of  the  planets  was  begun  by  Euler,  whose 
memoirs  on  the  mutual  perturbations  of  Jupiter  and  Saturn  gained  the  prizes 
of  the  French  Academy  in  1748  and  1752.  In  these  memoirs  was  given  the 
first  analytical  development  of  the  method  of  the  variation  of  parameters. 
The  equations  were  not  entirely  general  as  he  had  not  considered  the  elements 
as  being  all  simultaneously  variables.  The  first  steps  in  the  development  of 
the  perturbative  function  were  also  given  by  Euler. 

Lagrange,  whose  contributions  to  Celestial  Mechanics  were  of  the  most 
brilliant  character,  wrote  his  first  memoir  in  1766  on  the  perturbations  of 
Jupiter  and  Saturn.  In  this  work  he  developed  still  further  the  method  of 
the  variation  of  parameters,  leaving  his  final  equations,  however,  still  incorrect 
by  regarding  the  major  axes  and  the  epochs  of  the  perihelion  passages  as 
constants  in  deriving  the  equations  for  the  variations.  The  equations  for 
the  inclination,  node,  and  longitude  of  the  perihelion  from  the  node  were 
perfectly  correct.  In  the  expressions  for  the  mean  longitudes  of  the  planets 
there  were  terms  proportional  to  the  first  and  second  powers  of  the  time. 
These  were  entirely  due  to  the  imperfections  of  the  method,  their  true  form 
being  that  of  the  long  period  terms,  as  was  shown  by  Laplace  in  1784  by 
considering  terms  of  the  third  order  in  the  eccentricities.  The  method  of  the 
variation  of  parameters  was  completely  developed  for  the  first  time  in  1782 
by  Lagrange  in  a  prize  memoir  on  the  perturbations  of  comets  moving  in 


432  HISTORICAL   SKETCH. 

elliptical  orbits.  By  far  the  most  extensive  use  of  the  method  of  variation  of 
parameters  is  due  to  Delaunay,  whose  Lunar  Theory  is  essentially  a  long 
succession  of  the  applications  of  the  process,  each  step  of  it  removing  a  term 
from  the  perturbative  function. 

In  1773  Laplace  presented  his  first  memoir  to  the  French  Academy  of 
Sciences.  In  it  he  proved  his  celebrated  theorem  that,  up  to  the  second 
powers  of  the  eccentricities,  the  major  axes,  and  consequently  the  mean 
motions  of  the  planets,  have  no  secular  terms.  This  theorem  was  extended 
by  Lagrange  in  1774  and  1776  to  all  powers  of  the  eccentricities  and  of  the  sine 
of  the  angle  of  the  mutual  inclination,  for  perturbations  of  the  first  order  with 
respect  to  the  masses.  Poisson  proved  in  1809  that  the  major  axes  have  no 
purely  secular  terms  in  the  perturbations  of  the  second  order  with  respect  to 
the  masses.  Haretu  proved  in  his  Dissertation  at  Sorbonne  in  1878  that 
there  are  secular  variations  in  the  expressions  for  the  major  axes  in  the  terms 
of  the  third  order  with  respect  to  the  masses.  In  vol.  xix.  of  Annales  de 
I'Observatoire  de  Paris,  Eginitis  considered  terms  of  still  higher  order  with 
respect  to  the  masses. 

Lagrange  began  the  study  of  the  secular  terms  in  1774,  introducing  the 
variables  h,  I,  p,  and  q.  The  investigations  were  carried  on  by  Lagrange 
and  Laplace,  each  supplementing  and  extending  the  work  of  the  other,  until 
1784  when  their  work  became  complete  by  Laplace's  discovery  of  his  celebrated 
equations 

C, 


rf  rajttj-a/2  tan2  ij  =  C'. 
I 

These  equations  were  derived  by  using  only  the  linear  terms  in  the  differential 
equations.  Leverrier,  Hill,  and  others  have  extended  the  work  by  methods  of 
successive  approximations  to  terms  of  higher  degree.  Newcomb  (Smithsonian 
Contributions  to  Science,  vol.  xxi.,  1876)  has  established  the  more  far-reaching 
results  that  it  is  possible,  in  the  case  of  the  planetary  perturbations,  to  repre- 
sent the  elements  by  purely  periodic  functions  of  the  time  which  formally 
satisfy  the  differential  equations  of  motion.  If  these  series  were  convergent 
the  stability  of  the  solar  system  would  be  assured;  but  Poincare  has  shown 
that  they  are  in  general  divergent  (Les  Methodes  Nouvelles  de  la  Mecanique 
Celeste,  chap.  ix.).  Lindstedt  and  Gylden  have  also  succeeded  in  integrating 
the  equations  of  the  motion  of  n  bodies  in  periodic  series,  which,  however, 
are  in  general  divergent. 

Gauss,  Airy,  Adams,  Leverrier,  Hansen,  and  many  others  have  made 
important  contributions  to  the  planetary  theory  in  some  of  its  many  aspects. 
Adams  and  Leverrier  are  noteworthy  for  having  predicted  the  existence  and 
apparent  position  of  Neptune  from  the  unexplained  irregularities  in  the  motion 
of  Uranus.  More  recently  Poincare"  turned  his  attention  to  Celestial  Mechanics, 
publishing  a  prize  memoir  in  the  Ada  Mathematica,  vol.  xm.  This  memoir 
was  enlarged  and  published  in  book  form  with  the  title  Les  Methodes  Nouvelles 
de  la  Mecanique  Celeste.  Poincare"  applied  to  the  problem  all  the  resources 
of  modern  mathematics  with  unrivaled  genius;  he  brought  into  the  investiga- 
tion such  a  wealth  of  ideas,  and  he  devised  methods  of  such  immense  power 


HISTORICAL  SKETCH.  433 

that  the  subject  in  its  theoretical  aspects  has  been  entirely  revolutionized  in 
his  hands.  It  cannot  be  doubted  that  much  of  the  work  of  the  next  fifty 
years  will  be  in  amplifying  and  applying  the  processes  which  he  explained. 

The  following  works  should  be  consulted : 

Laplace's  Mecanique  Celeste,  containing  practically  all  that  was  known  of 
Celestial  Mechanics  at  the  time  it  was  written  (1799-1805). 

On  the  variation  of  parameters — Annales  de  V  Observatoire  de  Paris,  vol.  i. ; 
Tisserand's  Mecanique  Celeste,  vol.  i.;  Brown's  Lunar  Theory;  Dziobek's 
Planeten-Be  wegungen . 

On  the  development  of  the  perturbative  function — Annales  de  I 'Observatoire 
de  Paris,  vol.  i.;  Tisserand's  Mecanique  Celeste,  vol.  i.;  Hansen's  Entwickelung 
des  Products  einer  Potenz  des  Radius-Vectors  mil  dem  Sinus  oder  Cosinus  eines 
Vielfachen  der  wahren  Anomalie,  etc.,  Abh.  d.  K.  Sachs.  Ges.  zu  Leipzig,  vol.  n.; 
Newcomb's  memoir  on  the  General  Integrals  of  Planetary  Motion;  Poincare, 
Les  Methodes  Nouvelles,  vol.  i.,  chap.  vi. 

On  the  stability  of  the  solar  system — Tisserand's  Mecanique  Celeste,  vol.  i., 
chaps.  XL,  xxv.,  xxvi.,  and  vol.  iv.,  chap,  xxvi.;  Gylden,  Traite  Analytique 
des  Orbites  absolues,  vol.  i.;  Newcomb,  Smithsonian  Cont.,  vol.  xxi.;  Poincare, 
Les  Methodes  Nouvelles  de  la  Mecanique  Celeste,  vol.  n.,  chap.  x. 

On  the  subject  of  Celestial  Mechanics  as  a  whole  there  is  no  better  work 
available  than  that  of  Tisserand,  which  should  be  in  the  possession  of  every 
one  giving  special  attention  to  this  subject.  Another  noteworthy  work  is 
Charlier's  Mechanik  des  Himmels,  which,  besides  maintaining  a  high  order  of 
general  excellence,  is  unequaled  by  other  treatises  in  its  discussion  of  periodic 
solutions  of  the  Problem  of  Three  Bodies. 


29 


INDEX. 


Abbott,  66 

Acceleration  in  rectilinear  motion,  9 
curvilinear  motion,  1 1 
Adams,  363,  432 
Airy,  363,  365,  432 
Albategnius,  32 
Allegret,  319 
Almagest,  32 
Al-Sufi,  32 
Anaximander,  31 
Annual  equation,  348 
Anomaly,  eccentric,  159 

mean,  159 

true,  155 

Appell,  7,  10,  35,  97,  162 
Archimedes,  33 
Areal  velocity,  15 
Argument  of  latitude,  162 
Aristarchus,  31 
Aristotle,  31 

Atmospheres,  escape  of,  46 
Attraction  of  circular  discs,  103 

ellipsoids  99,  122,  127 
spheres,  99,   101,    104, 

114 
spheroids,  119,  132;  133 

Backhouse,  305 
Ball,  W.  W.  R.,  35 
Baltzer,  376 
Barker's  tables,  156 
Barnard,  305 
Bauschinger,  260 
Bernouilli,  Daniel,  190 

J.,  67 
Berry,  35 
Bertrand,  97 
Boltzmann,  3,  67 
Bour,  319 
Brorsen,  305 

Brown,  351,  352,  365,  433 
Bruns,  218,  276,  281 
Buchanan,  Daniel,  320 
Buchholz,  222,  260 
Buck,  320 
Budde,  35 
Burbury,  67 
Burnham,  85 

Cajori,  35 

Calory,  60 

Canonical  equations,  390 

Cantor,  35 


Carmichael,  35 
Cauchy,  367,  378 
Center  of  gravity,  22 

mass,  19,  20,  24 
Central  force,  69 
Chaldaeans,  31 
Chamberlin  and  Salisbury,  68 
Charlier,  216,  259,  427,  433 
Chasles,  138,  139 
Chauvenet,  190,  197 
Circular  orbits  for  three  bodies,  309 
Clairaut,  356,  363,  364,  367,  429,  431 
Clausius,  67 

Contraction  theory  of  sun's  heat,  63 
Copernicus,  33 

d'Alembert,  3,  7,  363,  429,  431 
Damoiseau,  364 
Darboux,  97,  138 

Darwin,  68,  139,  280,  281,  305,  320 
Delambre,  35 
Delaunay,  364,  430,  432 
De  Pontecoulant,  364 
Descartes,  190 
Despeyrous,  97,  138 
Differential  corrections,  162,  220 
Differential  equations  of  orbit,  80 
Dirichlet,  138 

Disturbing  forces,  resolution  of,  324 
Doolittle,  Eric,  361 
Double  points  of  surfaces  of  zero  ve- 
locity, 290 

Double  star  orbits,  85 
Duhring,  35 
Dziobek,  433 

Eccentric  Anomaly,  159 
Eginitis,  432 
Egyptians,  30 

Elements  of  orbits,  146,  148,  183 
Elements,  intermedisrte,  192 
Energy,  kinetic,  potential,  59 
Equations  of  relative  motion,  142 
Equipotential  curves,  283 

surfaces,  113 
Eratosthenes,  31 
Escape  of  atmospheres,  46 
Euclid,  32 
Euler,  24,  34,  138,  158,  190,  258,  363, 

364,  367,  429,  431 
Euler's  equation,  157,  275 
Evection,  the,  359 


434 


INDEX. 


435 


Falling  bodies,  36 

Force  varying  as  distance,  90 

inversely  as  square  of 
distance,  92 
fifth  power 
of  dis- 
tance, 93 

Galileo,  3,  33,  34,  67 
Gauss,  138,  139,  153,  154,  188,  190, 
193,  194,  231,  238,  240,  242,  243, 
244,  249,  259,  260,  360,  361,  432 
Gauss'  equations,  238,  240 
Gegenschein,  305 
Gibbs,  260 
Glaisher,  97 
Grant,  35 

Greek  philosophers,  30,  429 
Green,  109,  138,  139 
Griffin,  88,  97,  320 
Gylden,  305,  432 

Halley,  258,  348,  363 

Halphen,  97 

Hamilton,  3,  275 

Hankel,  35 

Hansen,  364,  430,  432,  433 

Haretu,  432 

Harkness  and  Morley,  292 

Harzer,  231,  232,  259 

Heat  of  sun,  59 

Height  of  projection,  45 

Helmholtz,  63,  68 

Herodotus,  30 

Herschel,  John,  325,  365 

William,  85 
Hertz,  3,  35 
Hilbert,  67 
Hill,  68,  280,  281,  287,  319,  351,  352, 

356,  361,  365,  430,  432 
Hipparchus,  31,  32,  359 
Holmes,  68 
Homoeoid,  100 
Huyghens,  34 

Ideler,  35 

Independent  star-numbers,  194 
Infinitesimal  body,  277 
Integrals  of  areas,  144,  264 

center  of  mass  141,  262 
Integral  of  energy,  267 
Integration  in  series,  172,  200,  202, 

227,  377 

Invariable  plane,  266 
Ivory,  116,  127,  132,  138 

Jacobi,  139,  267,  274,  275,  280,  281, 

319 

Jacobi's  integral,  280 
Jeans,  67 
Joule,  60 


Kepler,  33,  82,  83,  152,  190,  429 
Kepler's  equation,  159,  160,  163,  165 

laws,  82 

third  law,  152 
Kinetic  theory  of  gases,  46 
Kirchhoff,  3 
Klinkerfues,  222,  260 
Koenigs,  35,  97 

Laertius,  30 

Lagrange,  7,  34,  107,  132,  138,  161, 

193,  227,  259,  277,  312,  319,  363, 

364,  387,  418,  421,  423,  429,  431, 

432 
Lagrange's  brackets,  387 

quintic  equation,  312 
Lagrangian  solutions  of  the  problem 

of  three  bodies,  277,  291,  309,  313 
Lambert,  158,  258,  259 
Lane,  68 
Laplace,  34,  132,  138,  172,  193,  194, 

231,  249,  258,  259,  266,  275,  319, 

348,  350,  352,  362,  364,  367,  418, 

425,  429,  431,  432,  433 
Laue,  35 
Law  of  areas,  69 

converse  of,  73 
force  in  binary  stars,  86 
Laws  of  angular  and  linear  velocity,  73 
Kepler,  82 
motion,  3 
Lebon,  35 
Legendre,  97,  138 
Lehmann-Filhes,  319 
Leibnitz,  429 
Leonardo  da  Vinci,  33 
Leuschner,  222,  231,  232,  259 
Level  surfaces,  113 
Leverrier,    361,    363,  400,  406,  413, 

430,  432 
Levi-Civita,  268 
Linstedt,  319,  432 
Liouville,  319 
Long    period    inequalities  361,  371, 

416 

Longley,  320 
Love,  35 
Lubbock,  364 
Lunar  theory,  337 

MacCullagh,  138 
Mach,  3,  6,  35 
Maclaurin,  34,  132,  139 
MacMillan,  169,  320 
Marie,  35 
Mathieu,  319 
Maxwell,  67 
Mayer,  Robert,  68 
Tobias,  364 
McCormaok,  35 
Mean  anomaly,  159 


436 


INDEX. 


Mechanical  quadratures,  425 

Meteoric  theory  of  sun's  heat,  62 

Meton,  31 

Metonic  cycle,  31 

Meyer,  O.  E.,  67 

Motion  of  apsides,  352 

center  of  mass,  141,  262 
falling  particles,  36 

Neumann,  139 

Newcomb,  275,  361,  430,  432,  433 

Newton,  H.  A.,  62,  305 

Newton,  3,  5,  6,  7,  29,  33,  34,  67,  82, 
84,  97,  99,  101,  138,  190,  258,  275, 
320,  327,  350,  356,  365,  429,  431 

Newton's  law  of  gravitation,  82,  84 
laws  of  motion,  3 

Normal  form  of  differential  equations, 
75 

Node,  ascending,  descending,  182 

Nyren,  318 

Gibers,  259 

Omar,  32 

Oppolzer,  156,  222,  242,  260,  370 

Order  of  differential  equations,  74 

Osculating  conic,  322 

Parabolic  motion,  56 

Parallactic  inequality,  352 

Parallelogram  of  forces,  5 

Periodic  variations,  371,  413 

Perturbations,  meaning  of,  321 

by  oblate  body,  333 
resisting  medium,  333 
of  apsides,  352 
elements,  322,  382 
first  order,  382 
inclination,  343 
major  axis,  346 
node,  342 
period,  348 

Perturbative  function,  272 

resolution  of,  337, 

338,  345,  402 
development  of,  406 

Peurbach,  32 

Picard,  378,  428 

Plana,  364 

Planck,  35 

Plummer,  302 

Poincare,  35,  139,  267,  268,  275,  276, 
281,  320,  367,  378,  432,  433 

Poisson,  6,  138,  371,  418,  420,  432 

Poisspn  terms,  371 

Position  in  elliptic  orbits,  158 

hyperbolic  orbits,  177 
parabolic  orbits,  155 

Potential,  109,  261 

Precession  of  equinoxes,  344 

Preston,  60 


Problem  of  two  bodies,  140 
three  bodies,  277 
n  bodies,  261 

Ptolemy,  32,  359 

Pythagoras,  31 

Question  of  new  integrals,  268 

Radau,  274,  319 

Ratios  of  triangles,  233,  237 

Rectilinear  motion,  36 

Regiomontanus,  32 

Regions  of  real  and  imaginary  ve- 
locity, 286 

Relativity,  principle  of,  4 

Resolution  of  disturbing  force,  337, 
338 

Risteen,  67 

Ritter,  68 

Rodriguez,  138 

Routh,  35,  139 

Rowland,  60 

Rutherford,  68 

Salmon,  88 

Saracens,  32 

Saros,  31 

Secular  acceleration  of  moon's  motion, 
348 

Secular  variations,  360,  371,  417 

Solid  angles,  98 

Solution  of  linear  equations  by  ex- 
ponentials, 41 

Solutions  of  problem  of  three  bodies, 
290,  309,  313 

Speed,  8 

Spencer,  59 

Stability  of  solutions,  298,  306 

Stader,  97 

Stevinus,  33,  67 

Stirling,  138 

Stoney,  46 

Sturm,  139 

Surfaces  of  zero  relative  velocity,  281 

Sliter,  35 

Tait,  35 

Tait  and  Steele,  35,  97 

Tannery,  35 

Temperature  of  meteors,  61 

Thales,  30,  31 

Thomson,  139 

Thomson  and  Tait,  3,  104,  139,  283 

Time  aberration,  226 

Tisserand,  97,  139,  190,  260,  267,  276, 

295,  296,  312,  319,  365,  391,  407, 

426,  427,  433 
Tisserand's  criterion  for  identity  of 

comets,  295 
True  anomaly,  155 
Tycho  Brahe,  33,  348,  350 


INDEX.  437 

Uniform  motion,  8  Vis  viva  integral,  78,  267 

Ulugh  Beigh,  32  Voltaire,  190 
Units,  153 

canonical,  154  Waltherus,  32 

Variation,  the,  350  Waterson,  162 

Variation  of  coordinates,  321  Watson,  156,  242,  260 

elements,  322  Weierstrass,  367 

parameters  50,  322  Whewell,  35 

Vector,  5  Williamson,  161 

Velocity,  8  Wolf,  35 

areal,  15  Woodward,  4 

from  infinity,  45,  46  Work,  59 
of  escape,  48 

Villarceau,  259  Young,  164 


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